P. 356 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bnj1118 35501*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))    &   ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   ⊢ 𝐷 = (ω ∖ {∅})    &   ⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))    &   ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))    &   ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))    ⇒   ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)
Theorem	bnj1121 35502	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))    &   ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))    &   ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))    &   ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))    &   ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 𝜂)    &   ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}    ⇒   ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
Theorem	bnj1123 35503*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}    &   ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))    &   ⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂)    ⇒   ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
Theorem	bnj1030 35504*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))    &   ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))    &   ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))    &   ⊢ 𝐷 = (ω ∖ {∅})    &   ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}    &   ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))    &   ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))    &   ⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑)    &   ⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓)    &   ⊢ (𝜒′ ↔ [𝑗 / 𝑖]𝜒)    &   ⊢ (𝜃′ ↔ [𝑗 / 𝑖]𝜃)    &   ⊢ (𝜏′ ↔ [𝑗 / 𝑖]𝜏)    &   ⊢ (𝜁′ ↔ [𝑗 / 𝑖]𝜁)    &   ⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂)    &   ⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))    &   ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))    ⇒   ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
Theorem	bnj1124 35505	Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))    &   ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    ⇒   ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
Theorem	bnj1133 35506*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐷 = (ω ∖ {∅})    &   ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))    &   ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))    &   ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)    ⇒   ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃)
Theorem	bnj1128 35507*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ⊢ 𝐷 = (ω ∖ {∅})    &   ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}    &   ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))    &   ⊢ (𝜃 ↔ (𝜒 → (𝑓‘𝑖) ⊆ 𝐴))    &   ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))    &   ⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑)    &   ⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓)    &   ⊢ (𝜒′ ↔ [𝑗 / 𝑖]𝜒)    &   ⊢ (𝜃′ ↔ [𝑗 / 𝑖]𝜃)    ⇒   ⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌 ∈ 𝐴)
Theorem	bnj1127 35508	Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌 ∈ 𝐴)
Theorem	bnj1125 35509	Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
Theorem	bnj1145 35510*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ⊢ 𝐷 = (ω ∖ {∅})    &   ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}    &   ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))    &   ⊢ (𝜃 ↔ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)))    ⇒   ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
Theorem	bnj1147 35511	Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
Theorem	bnj1137 35512*	Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo(𝐵, 𝐴, 𝑅))
Theorem	bnj1148 35513	Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)
Theorem	bnj1136 35514*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    &   ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))    &   ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)
Theorem	bnj1152 35515	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))
Theorem	bnj1154 35516*	Property of Fr. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
Theorem	bnj1171 35517	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)    &   ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))    ⇒   ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
Theorem	bnj1172 35518	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝜃 ↔ 𝑤 ∈ 𝐴))    ⇒   ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
Theorem	bnj1173 35519	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝜃 ↔ 𝑤 ∈ 𝐴))
Theorem	bnj1174 35520	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶))))    &   ⊢ (𝜃 → (𝑤𝑅𝑧 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))    ⇒   ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
Theorem	bnj1175 35521	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   ⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧)))    &   ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴))    ⇒   ⊢ (𝜃 → (𝑤𝑅𝑧 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))
Theorem	bnj1176 35522*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V))    &   ⊢ ((𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V) → ∃𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐶 ¬ 𝑤𝑅𝑧)    ⇒   ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶))))
Theorem	bnj1177 35523	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 ↔ (𝑋 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑋))    &   ⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V))
Theorem	bnj1186 35524*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧)
Theorem	bnj1190 35525*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅))    &   ⊢ (𝜓 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑥))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ∃𝑤 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑤)
Theorem	bnj1189 35526*	Technical lemma for bnj69 35527. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅))    &   ⊢ (𝜓 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑥))    &   ⊢ (𝜒 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
21.4.3  The existence of a minimal element in certain classes
Theorem	bnj69 35527*	Existence of a minimal element in certain classes: if 𝑅 is well-founded and set-like on 𝐴, then every nonempty subclass of 𝐴 has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
Theorem	bnj1228 35528*	Existence of a minimal element in certain classes: if 𝑅 is well-founded and set-like on 𝐴, then every nonempty subclass of 𝐴 has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
21.4.4  Well-founded induction
Theorem	bnj1204 35529*	Well-founded induction. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑))    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	bnj1234 35530*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐷 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))}    ⇒   ⊢ 𝐶 = 𝐷
Theorem	bnj1245 35531*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))    &   ⊢ 𝑍 = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐾 = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑍))}    ⇒   ⊢ (𝜑 → dom ℎ ⊆ 𝐴)
Theorem	bnj1256 35532*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑)
Theorem	bnj1259 35533*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)
Theorem	bnj1253 35534*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))    ⇒   ⊢ (𝜑 → 𝐸 ≠ ∅)
Theorem	bnj1279 35535*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))    ⇒   ⊢ ((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
Theorem	bnj1286 35536*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))    ⇒   ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)
Theorem	bnj1280 35537*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜓 → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)    ⇒   ⊢ (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅)))
Theorem	bnj1296 35538*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅)))    &   ⊢ 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐾 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))}    &   ⊢ 𝑊 = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐿 = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑊))}    ⇒   ⊢ (𝜓 → (𝑔‘𝑥) = (ℎ‘𝑥))
Theorem	bnj1309 35539*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    ⇒   ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
Theorem	bnj1307 35540*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)    ⇒   ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)
Theorem	bnj1311 35541*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))
Theorem	bnj1318 35542	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑋 = 𝑌 → trCl(𝑋, 𝐴, 𝑅) = trCl(𝑌, 𝐴, 𝑅))
Theorem	bnj1326 35543*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))
Theorem	bnj1321 35544*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)
Theorem	bnj1364 35545	Property of FrSe. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑅 FrSe 𝐴 → 𝑅 Se 𝐴)
Theorem	bnj1371 35546*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))    ⇒   ⊢ (𝑓 ∈ 𝐻 → Fun 𝑓)
Theorem	bnj1373 35547*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    ⇒   ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
Theorem	bnj1374 35548*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    ⇒   ⊢ (𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶)
Theorem	bnj1384 35549*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    ⇒   ⊢ (𝑅 FrSe 𝐴 → Fun 𝑃)
Theorem	bnj1388 35550*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    ⇒   ⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
Theorem	bnj1398 35551*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))    &   ⊢ (𝜂 ↔ (𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))    ⇒   ⊢ (𝜒 → ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃)
Theorem	bnj1413 35552*	Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐵 ∈ V)
Theorem	bnj1408 35553*	Technical lemma for bnj1414 35554. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    &   ⊢ 𝐶 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    &   ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))    &   ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)
Theorem	bnj1414 35554*	Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)
Theorem	bnj1415 35555*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    ⇒   ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
Theorem	bnj1416 35556	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))    ⇒   ⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
Theorem	bnj1418 35557	Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
Theorem	bnj1417 35558*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
⊢ (𝜑 ↔ 𝑅 FrSe 𝐴)    &   ⊢ (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))    &   ⊢ (𝜃 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒))    &   ⊢ 𝐵 = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
Theorem	bnj1421 35559*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ (𝜒 → Fun 𝑃)    &   ⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))    ⇒   ⊢ (𝜒 → Fun 𝑄)
Theorem	bnj1444 35560*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))    &   ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))    &   ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))    &   ⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))    ⇒   ⊢ (𝜌 → ∀𝑦𝜌)
Theorem	bnj1445 35561*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))    &   ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))    &   ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))    &   ⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))    &   ⊢ (𝜎 ↔ (𝜌 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))    &   ⊢ (𝜑 ↔ (𝜎 ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))    &   ⊢ 𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩    ⇒   ⊢ (𝜎 → ∀𝑑𝜎)
Theorem	bnj1446 35562*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    ⇒   ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑑(𝑄‘𝑧) = (𝐺‘𝑊))
Theorem	bnj1447 35563*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    ⇒   ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑦(𝑄‘𝑧) = (𝐺‘𝑊))
Theorem	bnj1448 35564*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    ⇒   ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑓(𝑄‘𝑧) = (𝐺‘𝑊))
Theorem	bnj1449 35565*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))    &   ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))    &   ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))    ⇒   ⊢ (𝜁 → ∀𝑓𝜁)
Theorem	bnj1442 35566*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))    &   ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))    ⇒   ⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊))
Theorem	bnj1450 35567*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))    &   ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))    &   ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))    &   ⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))    &   ⊢ (𝜎 ↔ (𝜌 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))    &   ⊢ (𝜑 ↔ (𝜎 ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))    &   ⊢ 𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩    ⇒   ⊢ (𝜁 → (𝑄‘𝑧) = (𝐺‘𝑊))
Theorem	bnj1423 35568*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    ⇒   ⊢ (𝜒 → ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))
Theorem	bnj1452 35569*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    ⇒   ⊢ (𝜒 → 𝐸 ∈ 𝐵)
Theorem	bnj1466 35570*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    ⇒   ⊢ (𝑤 ∈ 𝑄 → ∀𝑓 𝑤 ∈ 𝑄)
Theorem	bnj1467 35571*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    ⇒   ⊢ (𝑤 ∈ 𝑄 → ∀𝑑 𝑤 ∈ 𝑄)
Theorem	bnj1463 35572*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   ⊢ (𝜒 → 𝑄 ∈ V)    &   ⊢ (𝜒 → ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))    &   ⊢ (𝜒 → 𝑄 Fn 𝐸)    &   ⊢ (𝜒 → 𝐸 ∈ 𝐵)    ⇒   ⊢ (𝜒 → 𝑄 ∈ 𝐶)
Theorem	bnj1489 35573*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    ⇒   ⊢ (𝜒 → 𝑄 ∈ V)
Theorem	bnj1491 35574*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ (𝜒 → (𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    ⇒   ⊢ ((𝜒 ∧ 𝑄 ∈ V) → ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
Theorem	bnj1312 35575*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e., a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}    &   ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))    &   ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)    &   ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   ⊢ 𝑃 = ∪ 𝐻    &   ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})    &   ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    ⇒   ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
Theorem	bnj1493 35576*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    ⇒   ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
Theorem	bnj1497 35577*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    ⇒   ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔
Theorem	bnj1498 35578*	Technical lemma for bnj60 35579. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
21.4.5  Well-founded recursion, part 1 of 3
Theorem	bnj60 35579*	Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)
Theorem	bnj1514 35580*	Technical lemma for bnj1500 35585. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    ⇒   ⊢ (𝑓 ∈ 𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
Theorem	bnj1518 35581*	Technical lemma for bnj1500 35585. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))    ⇒   ⊢ (𝜓 → ∀𝑑𝜓)
Theorem	bnj1519 35582*	Technical lemma for bnj1500 35585. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Theorem	bnj1520 35583*	Technical lemma for bnj1500 35585. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Theorem	bnj1501 35584*	Technical lemma for bnj1500 35585. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑))    ⇒   ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
21.4.6  Well-founded recursion, part 2 of 3
Theorem	bnj1500 35585*	Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Theorem	bnj1525 35586*	Technical lemma for bnj1522 35589. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝐹 ≠ 𝐻))    ⇒   ⊢ (𝜓 → ∀𝑥𝜓)
Theorem	bnj1529 35587*	Technical lemma for bnj1522 35589. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))    &   ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)    ⇒   ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
Theorem	bnj1523 35588*	Technical lemma for bnj1522 35589. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝐹 ≠ 𝐻))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ≠ (𝐻‘𝑥)))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)}    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑦 ∈ 𝐷 ∧ ∀𝑧 ∈ 𝐷 ¬ 𝑧𝑅𝑦))    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)
21.4.7  Well-founded recursion, part 3 of 3
Theorem	bnj1522 35589*	Well-founded recursion, part 3 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)
21.5  Mathbox for BTernaryTau
21.5.1  First-order logic
21.5.1.1  Auxiliary axiom schemes
Theorem	nfan1c 35590	Variant of nfan 1932 and commuted form of nfan1 2238. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ Ⅎ𝑥(𝜓 ∧ 𝜑)
Theorem	cbvex1v 35591*	Rule used to change bound variables, using implicit substitution. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑦𝜒))
Theorem	dvelimalcased 35592*	Eliminate a disjoint variable condition from a universally quantified statement using cases. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜓 → 𝜃)))    &   ⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (𝜒 → 𝜃))    &   ⊢ (𝜑 → ∀𝑧𝜓)    &   ⊢ (𝜑 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → ∀𝑥𝜃)
Theorem	dvelimalcasei 35593*	Eliminate a disjoint variable condition from a universally quantified statement using cases. Inference form of dvelimalcased 35592. See axsepg2 35674 for an example of its use. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒)    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑 → 𝜒)))    &   ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒))    &   ⊢ ∀𝑧𝜑    &   ⊢ ∀𝑥𝜓    ⇒   ⊢ ∀𝑥𝜒
Theorem	dvelimexcased 35594*	Eliminate a disjoint variable condition from an existentially quantified statement using cases. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜓 → 𝜃)))    &   ⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (𝜒 → 𝜃))    &   ⊢ (𝜑 → ∃𝑧𝜓)    &   ⊢ (𝜑 → ∃𝑥𝜒)    ⇒   ⊢ (𝜑 → ∃𝑥𝜃)
Theorem	dvelimexcasei 35595*	Eliminate a disjoint variable condition from an existentially quantified statement using cases. Inference form of dvelimexcased 35594. See axnulg 35679 for an example of its use. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒)    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑 → 𝜒)))    &   ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒))    &   ⊢ ∃𝑧𝜑    &   ⊢ ∃𝑥𝜓    ⇒   ⊢ ∃𝑥𝜒
21.5.2  ZF set theory
Theorem	inv2 35596	The intersection of the universal class with a class is itself. A commuted form of inv1 4351. (Contributed by BTernaryTau, 24-Jun-2026.)
⊢ (V ∩ 𝐴) = 𝐴
Theorem	exdifsn 35597	There exists an element in a class excluding a singleton if and only if there exists an element in the original class not equal to the singleton element. (Contributed by BTernaryTau, 15-Sep-2023.)
⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 𝐵)
Theorem	axnulALT2 35598*	Alternate proof of axnul 5266, proved from propositional calculus, ax-gen 1828, ax-4 1842, ax-6 2000, and ax-rep 5236. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BTernaryTau, 27-Mar-2026.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	fnfvintima 35599*	Condition for a function value to equal the intersection of an image that contains it. (Contributed by BTernaryTau, 23-Jun-2026.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝐶) = ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥)))
Theorem	ordprcon 35600	If an ordinal class is not a set, then it must be the proper class of all ordinals. (Contributed by BTernaryTau, 9-Jun-2026.)
⊢ ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → 𝐴 = On)