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Theorem List for Metamath Proof Explorer - 31501-31600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembnj1352 31501* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theorembnj1361 31502* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)

Theorembnj1366 31503* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
(𝜓 ↔ (𝐴 ∈ V ∧ ∀𝑥𝐴 ∃!𝑦𝜑𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑}))       (𝜓𝐵 ∈ V)

Theorembnj1379 31504* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)    &   𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))    &   (𝜒 ↔ (𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))    &   (𝜃 ↔ (𝜒𝑓𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓))    &   (𝜏 ↔ (𝜃𝑔𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑔))       (𝜓 → Fun 𝐴)

Theorembnj1383 31505* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)    &   𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))       (𝜓 → Fun 𝐴)

Theorembnj1385 31506* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)    &   𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))    &   (𝑥𝐴 → ∀𝑓 𝑥𝐴)    &   (𝜑′ ↔ ∀𝐴 Fun )    &   𝐸 = (dom ∩ dom 𝑔)    &   (𝜓′ ↔ (𝜑′ ∧ ∀𝐴𝑔𝐴 (𝐸) = (𝑔𝐸)))       (𝜓 → Fun 𝐴)

Theorembnj1386 31507* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)    &   𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))    &   (𝑥𝐴 → ∀𝑓 𝑥𝐴)       (𝜓 → Fun 𝐴)

Theorembnj1397 31508 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜓 → ∀𝑥𝜓)       (𝜑𝜓)

Theorembnj1400 31509* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       dom 𝐴 = 𝑥𝐴 dom 𝑥

Theorembnj1405 31510* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝑋 𝑦𝐴 𝐵)       (𝜑 → ∃𝑦𝐴 𝑋𝐵)

Theorembnj1422 31511 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → Fun 𝐴)    &   (𝜑 → dom 𝐴 = 𝐵)       (𝜑𝐴 Fn 𝐵)

Theorembnj1424 31512 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       (𝐷𝐴 → (𝐷𝐵𝐷𝐶))

Theorembnj1436 31513 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝜑}       (𝑥𝐴𝜑)

Theorembnj1441 31514* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑥𝐴 → ∀𝑦 𝑥𝐴)    &   (𝜑 → ∀𝑦𝜑)       (𝑧 ∈ {𝑥𝐴𝜑} → ∀𝑦 𝑧 ∈ {𝑥𝐴𝜑})

Theorembnj1454 31515 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝜑}       (𝐵 ∈ V → (𝐵𝐴[𝐵 / 𝑥]𝜑))

Theorembnj1459 31516* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ (𝜑𝑥𝐴))    &   (𝜓𝜒)       (𝜑 → ∀𝑥𝐴 𝜒)

Theorembnj1464 31517* Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))

Theorembnj1465 31518* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒𝜓)       ((𝜒𝐴𝑉) → ∃𝑥𝜑)

Theorembnj1468 31519* Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))

Theorembnj1476 31520 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = {𝑥𝐴 ∣ ¬ 𝜑}    &   (𝜓𝐷 = ∅)       (𝜓 → ∀𝑥𝐴 𝜑)

Theorembnj1502 31521 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → Fun 𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑𝐴 ∈ dom 𝐺)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Theorembnj1503 31522 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → Fun 𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑𝐴 ⊆ dom 𝐺)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Theorembnj1517 31523 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥 ∣ (𝜑𝜓)}       (𝑥𝐴𝜓)

Theorembnj1521 31524 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∃𝑥𝐵 𝜑)    &   (𝜃 ↔ (𝜒𝑥𝐵𝜑))    &   (𝜒 → ∀𝑥𝜒)       (𝜒 → ∃𝑥𝜃)

Theorembnj1533 31525 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃 → ∀𝑧𝐵 ¬ 𝑧𝐷)    &   𝐵𝐴    &   𝐷 = {𝑧𝐴𝐶𝐸}       (𝜃 → ∀𝑧𝐵 𝐶 = 𝐸)

Theorembnj1534 31526* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}    &   (𝑤𝐹 → ∀𝑥 𝑤𝐹)       𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}

Theorembnj1536 31527* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))       (𝜑 → (𝐹𝐵) = (𝐺𝐵))

Theorembnj1538 31528 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝐵𝜑}       (𝑥𝐴𝜑)

Theorembnj1541 31529 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝐴𝐵))    &    ¬ 𝜑       (𝜓𝐴 = 𝐵)

Theorembnj1542 31530* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   (𝜑𝐹𝐺)    &   (𝑤𝐹 → ∀𝑥 𝑤𝐹)       (𝜑 → ∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥))

20.4.2  Well founded induction and recursion

Theorembnj110 31531* Well-founded induction restricted to a set (𝐴 ∈ V). 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.)
𝐴 ∈ V    &   (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))       ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)

Theorembnj157 31532* Well-founded induction restricted to a set (𝐴 ∈ V). 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 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))    &   𝐴 ∈ V    &   𝑅 Fr 𝐴       (∀𝑥𝐴 (𝜓𝜑) → ∀𝑥𝐴 𝜑)

Theorembnj66 31533* Technical lemma for bnj60 31733. 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 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}       (𝑔𝐶 → Rel 𝑔)

Theorembnj91 31534* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   𝑍 ∈ V       ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Theorembnj92 31535* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝑍 ∈ V       ([𝑍 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑍 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj93 31536* Technical lemma for bnj97 31539. 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.)
((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)

Theorembnj95 31537 Technical lemma for bnj124 31544. 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.)
𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}       𝐹 ∈ V

Theorembnj96 31538* Technical lemma for bnj150 31549. 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.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}       ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1o)

Theorembnj97 31539* Technical lemma for bnj150 31549. 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.)
𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}       ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))

Theorembnj98 31540 Technical lemma for bnj150 31549. 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.)
𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))

Theorembnj106 31541* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐹 ∈ V       ([𝐹 / 𝑓][1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj118 31542* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜑′[1o / 𝑛]𝜑)       (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Theorembnj121 31543* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜁′[1o / 𝑛]𝜁)    &   (𝜑′[1o / 𝑛]𝜑)    &   (𝜓′[1o / 𝑛]𝜓)       (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)))

Theorembnj124 31544* Technical lemma for bnj150 31549. 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.)
𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}    &   (𝜑″[𝐹 / 𝑓]𝜑′)    &   (𝜓″[𝐹 / 𝑓]𝜓′)    &   (𝜁″[𝐹 / 𝑓]𝜁′)    &   (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)))       (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″𝜓″)))

Theorembnj125 31545* Technical lemma for bnj150 31549. 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))    &   (𝜑′[1o / 𝑛]𝜑)    &   (𝜑″[𝐹 / 𝑓]𝜑′)    &   𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}       (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))

Theorembnj126 31546* Technical lemma for bnj150 31549. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓′[1o / 𝑛]𝜓)    &   (𝜓″[𝐹 / 𝑓]𝜓′)    &   𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}       (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj130 31547* Technical lemma for bnj151 31550. 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 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜑′[1o / 𝑛]𝜑)    &   (𝜓′[1o / 𝑛]𝜓)    &   (𝜃′[1o / 𝑛]𝜃)       (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1o𝜑′𝜓′)))

Theorembnj149 31548* Technical lemma for bnj151 31550. 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.)
(𝜃1 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1o𝜑′𝜓′)))    &   (𝜁0 ↔ (𝑓 Fn 1o𝜑′𝜓′))    &   (𝜁1[𝑔 / 𝑓]𝜁0)    &   (𝜑1[𝑔 / 𝑓]𝜑′)    &   (𝜓1[𝑔 / 𝑓]𝜓′)    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))       𝜃1

Theorembnj150 31549* Technical lemma for bnj151 31550. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜑′[1o / 𝑛]𝜑)    &   (𝜓′[1o / 𝑛]𝜓)    &   (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1o𝜑′𝜓′)))    &   (𝜁′[1o / 𝑛]𝜁)    &   𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}    &   (𝜑″[𝐹 / 𝑓]𝜑′)    &   (𝜓″[𝐹 / 𝑓]𝜓′)    &   (𝜁″[𝐹 / 𝑓]𝜁′)       𝜃0

Theorembnj151 31550* Technical lemma for bnj153 31553. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜏 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜃))    &   (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜑′[1o / 𝑛]𝜑)    &   (𝜓′[1o / 𝑛]𝜓)    &   (𝜃′[1o / 𝑛]𝜃)    &   (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1o𝜑′𝜓′)))    &   (𝜃1 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1o𝜑′𝜓′)))    &   (𝜁′[1o / 𝑛]𝜁)    &   𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}    &   (𝜑″[𝐹 / 𝑓]𝜑′)    &   (𝜓″[𝐹 / 𝑓]𝜓′)    &   (𝜁″[𝐹 / 𝑓]𝜁′)    &   (𝜁0 ↔ (𝑓 Fn 1o𝜑′𝜓′))    &   (𝜁1[𝑔 / 𝑓]𝜁0)    &   (𝜑1[𝑔 / 𝑓]𝜑′)    &   (𝜓1[𝑔 / 𝑓]𝜓′)       (𝑛 = 1o → ((𝑛𝐷𝜏) → 𝜃))

Theorembnj154 31551* Technical lemma for bnj153 31553. 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.)
(𝜑1[𝑔 / 𝑓]𝜑′)    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))       (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))

Theorembnj155 31552* Technical lemma for bnj153 31553. 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.)
(𝜓1[𝑔 / 𝑓]𝜓′)    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       (𝜓1 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj153 31553* Technical lemma for bnj852 31594. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜏 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜃))       (𝑛 = 1o → ((𝑛𝐷𝜏) → 𝜃))

Theorembnj207 31554* Technical lemma for bnj852 31594. 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 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜑′[𝑀 / 𝑛]𝜑)    &   (𝜓′[𝑀 / 𝑛]𝜓)    &   (𝜒′[𝑀 / 𝑛]𝜒)    &   𝑀 ∈ V       (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))

Theorembnj213 31555 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

Theorembnj222 31556* Technical lemma for bnj229 31557. 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 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))       (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))

Theorembnj229 31557* Technical lemma for bnj517 31558. 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 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) ⊆ 𝐴)

Theorembnj517 31558* Technical lemma for bnj518 31559. 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 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑁 ∈ ω ∧ 𝜑𝜓) → ∀𝑛𝑁 (𝐹𝑛) ⊆ 𝐴)

Theorembnj518 31559* Technical lemma for bnj852 31594. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛))       ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Theorembnj523 31560* Technical lemma for bnj852 31594. 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))    &   (𝜑′[𝑀 / 𝑛]𝜑)    &   𝑀 ∈ V       (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))

Theorembnj526 31561* Technical lemma for bnj852 31594. 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))    &   (𝜑″[𝐺 / 𝑓]𝜑)    &   𝐺 ∈ V       (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Theorembnj528 31562 Technical lemma for bnj852 31594. 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.)
𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})       𝐺 ∈ V

Theorembnj535 31563* Technical lemma for bnj852 31594. 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 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))       ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)

Theorembnj539 31564* Technical lemma for bnj852 31594. 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 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓′[𝑀 / 𝑛]𝜓)    &   𝑀 ∈ V       (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑀 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj540 31565* Technical lemma for bnj852 31594. 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 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   𝐺 ∈ V       (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj543 31566* Technical lemma for bnj852 31594. 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 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))       ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)

Theorembnj544 31567* Technical lemma for bnj852 31594. 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 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))       ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)

Theorembnj545 31568 Technical lemma for bnj852 31594. 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))    &   𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)    &   (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))       ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)

Theorembnj546 31569* Technical lemma for bnj852 31594. 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 ∖ {∅})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑅 FrSe 𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Theorembnj548 31570* Technical lemma for bnj852 31594. 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.)
(𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)       (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝐵 = 𝐾)

Theorembnj553 31571* Technical lemma for bnj852 31594. 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 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)       (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐿)

Theorembnj554 31572* Technical lemma for bnj852 31594. 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.)
(𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)       ((𝜂𝜁) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))

Theorembnj556 31573 Technical lemma for bnj852 31594. 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.)
(𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))       (𝜂𝜎)

Theorembnj557 31574* Technical lemma for bnj852 31594. 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 ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)       ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺𝑚) = 𝐿)

Theorembnj558 31575* Technical lemma for bnj852 31594. 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 ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)       ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺‘suc 𝑖) = 𝐾)

Theorembnj561 31576 Technical lemma for bnj852 31594. 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.)
(𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)       ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)

Theorembnj562 31577 Technical lemma for bnj852 31594. 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.)
(𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)       ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜑″)

Theorembnj570 31578* Technical lemma for bnj852 31594. 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 ∖ {∅})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = 𝐾)

Theorembnj571 31579* Technical lemma for bnj852 31594. 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 ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)    &   (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)    &   (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)

Theorembnj605 31580* Technical lemma. 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 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))    &   (𝜑″[𝑓 / 𝑓]𝜑)    &   (𝜓″[𝑓 / 𝑓]𝜓)    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   𝑓 ∈ V    &   (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))    &   (𝜑″ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝑝𝜂)    &   ((𝜃𝑚𝐷𝑚 E 𝑛) → 𝜒′)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝑓 Fn 𝑛)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜑″)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)       ((𝑛 ≠ 1o𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 𝑛𝜑𝜓)))

Theorembnj581 31581* Technical lemma for bnj580 31586. 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). (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑔 / 𝑓]𝜑)    &   (𝜓′[𝑔 / 𝑓]𝜓)    &   (𝜒′[𝑔 / 𝑓]𝜒)       (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))

Theorembnj589 31582* Technical lemma for bnj852 31594. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       (𝜓 ↔ ∀𝑘 ∈ ω (suc 𝑘𝑛 → (𝑓‘suc 𝑘) = 𝑦 ∈ (𝑓𝑘) pred(𝑦, 𝐴, 𝑅)))

Theorembnj590 31583 Technical lemma for bnj852 31594. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝐵 = suc 𝑖𝜓) → (𝑖 ∈ ω → (𝐵𝑛 → (𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))

Theorembnj591 31584* Technical lemma for bnj852 31594. 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 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))       ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))

Theorembnj594 31585* Technical lemma for bnj852 31594. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))    &   (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))    &   ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))    &   (𝜏 ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]𝜃))       ((𝑗𝑛𝜏) → 𝜃)

Theorembnj580 31586* Technical lemma for bnj579 31587. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑔 / 𝑓]𝜑)    &   (𝜓′[𝑔 / 𝑓]𝜓)    &   (𝜒′[𝑔 / 𝑓]𝜒)    &   𝐷 = (ω ∖ {∅})    &   (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))    &   (𝜏 ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]𝜃))       (𝑛𝐷 → ∃*𝑓𝜒)

Theorembnj579 31587* Technical lemma for bnj852 31594. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})       (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))

Theorembnj602 31588 Equality theorem for the pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))

Theorembnj607 31589* Technical lemma for bnj852 31594. 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 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))    &   (𝜑″[𝐺 / 𝑓]𝜑)    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   𝐺 ∈ V    &   (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))    &   (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝑝𝜂)    &   ((𝜃𝑚𝐷𝑚 E 𝑛) → 𝜒′)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜑″)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)    &   (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜑0[ / 𝑓]𝜑)    &   (𝜓0[ / 𝑓]𝜓)    &   (𝜑1[𝐺 / ]𝜑0)    &   (𝜓1[𝐺 / ]𝜓0)       ((𝑛 ≠ 1o𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 𝑛𝜑𝜓)))

Theorembnj609 31590* Technical lemma for bnj852 31594. 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))    &   (𝜑″[𝐺 / 𝑓]𝜑)    &   𝐺 ∈ V       (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Theorembnj611 31591* Technical lemma for bnj852 31594. 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 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   𝐺 ∈ V       (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj600 31592* Technical lemma for bnj852 31594. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))    &   (𝜑′[𝑚 / 𝑛]𝜑)    &   (𝜓′[𝑚 / 𝑛]𝜓)    &   (𝜒′[𝑚 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑)    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   (𝜒″[𝐺 / 𝑓]𝜒)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})       (𝑛 ≠ 1o → ((𝑛𝐷𝜃) → 𝜒))

Theorembnj601 31593* Technical lemma for bnj852 31594. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))       (𝑛 ≠ 1o → ((𝑛𝐷𝜃) → 𝜒))

Theorembnj852 31594* Technical lemma for bnj69 31681. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})       ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))

Theorembnj864 31595* Technical lemma for bnj69 31681. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))    &   (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))       (𝜒 → ∃!𝑓𝜃)

Theorembnj865 31596* Technical lemma for bnj69 31681. 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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))    &   (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))       𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃)

Theorembnj873 31597* Technical lemma for bnj69 31681. 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 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜑′[𝑔 / 𝑓]𝜑)    &   (𝜓′[𝑔 / 𝑓]𝜓)       𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}

Theorembnj849 31598* Technical lemma for bnj69 31681. 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 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))    &   (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑔 / 𝑓]𝜑)    &   (𝜓′[𝑔 / 𝑓]𝜓)    &   (𝜃′[𝑔 / 𝑓]𝜃)    &   (𝜏 ↔ (𝑅 FrSe 𝐴𝑋𝐴))       ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)

Theorembnj882 31599* Definition (using hypotheses for readability) of the function giving the transitive closure of 𝑋 in 𝐴 by 𝑅. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}        trCl(𝑋, 𝐴, 𝑅) = 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖)

Theorembnj18eq1 31600 Equality theorem for transitive closure. (Contributed by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
(𝑋 = 𝑌 → trCl(𝑋, 𝐴, 𝑅) = trCl(𝑌, 𝐴, 𝑅))

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