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Theorem List for Metamath Proof Explorer - 30001-30100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj1383 30001* 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 30002* 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 30003* 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 30004 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜓 → ∀𝑥𝜓)       (𝜑𝜓)
 
Theorembnj1400 30005* 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 30006* 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 30007 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 30008 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       (𝐷𝐴 → (𝐷𝐵𝐷𝐶))
 
Theorembnj1436 30009 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝜑}       (𝑥𝐴𝜑)
 
Theorembnj1441 30010* 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 30011 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 30012* 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 30013* 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 30014* 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 30015* 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 30016 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 30017 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 30018 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 30019 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥 ∣ (𝜑𝜓)}       (𝑥𝐴𝜓)
 
Theorembnj1521 30020 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 30021 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 30022* 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 30023* 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 30024 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝐵𝜑}       (𝑥𝐴𝜑)
 
Theorembnj1541 30025 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝐴𝐵))    &    ¬ 𝜑       (𝜓𝐴 = 𝐵)
 
Theorembnj1542 30026* 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 30027* 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 30028* 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 30029* Technical lemma for bnj60 30229. 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 30030* 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 30031* 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 30032* Technical lemma for bnj97 30035. 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 30033 Technical lemma for bnj124 30040. 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 30034* Technical lemma for bnj150 30045. 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 30035* Technical lemma for bnj150 30045. 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 30036 Technical lemma for bnj150 30045. 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 30037* 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 30038* 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 30039* 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 30040* Technical lemma for bnj150 30045. 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 30041* Technical lemma for bnj150 30045. 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 30042* Technical lemma for bnj150 30045. 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 30043* Technical lemma for bnj151 30046. 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 30044* Technical lemma for bnj151 30046. 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 30045* Technical lemma for bnj151 30046. 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 30046* Technical lemma for bnj153 30049. 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 30047* Technical lemma for bnj153 30049. 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 30048* Technical lemma for bnj153 30049. 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 30049* Technical lemma for bnj852 30090. 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 30050* Technical lemma for bnj852 30090. 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 30051 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
 
Theorembnj222 30052* Technical lemma for bnj229 30053. 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 30053* Technical lemma for bnj517 30054. 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 30054* Technical lemma for bnj518 30055. 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 30055* Technical lemma for bnj852 30090. 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 30056* Technical lemma for bnj852 30090. 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 30057* Technical lemma for bnj852 30090. 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 30058 Technical lemma for bnj852 30090. 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 30059* Technical lemma for bnj852 30090. 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 30060* Technical lemma for bnj852 30090. 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 30061* Technical lemma for bnj852 30090. 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 30062* Technical lemma for bnj852 30090. 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 30063* Technical lemma for bnj852 30090. 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 30064 Technical lemma for bnj852 30090. 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 30065* Technical lemma for bnj852 30090. 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 30066* Technical lemma for bnj852 30090. 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 30067* Technical lemma for bnj852 30090. 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 30068* Technical lemma for bnj852 30090. 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 30069 Technical lemma for bnj852 30090. 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 30070* Technical lemma for bnj852 30090. 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 30071* Technical lemma for bnj852 30090. 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 30072 Technical lemma for bnj852 30090. 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 30073 Technical lemma for bnj852 30090. 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 30074* Technical lemma for bnj852 30090. 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 30075* Technical lemma for bnj852 30090. 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 30076* 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 30077* Technical lemma for bnj580 30082. 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 30078* Technical lemma for bnj852 30090. 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 30079 Technical lemma for bnj852 30090. 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 30080* Technical lemma for bnj852 30090. 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 30081* Technical lemma for bnj852 30090. 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 30082* Technical lemma for bnj579 30083. 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 30083* Technical lemma for bnj852 30090. 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 30084 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 30085* Technical lemma for bnj852 30090. 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 30086* Technical lemma for bnj852 30090. 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 30087* Technical lemma for bnj852 30090. 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 30088* Technical lemma for bnj852 30090. 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 30089* Technical lemma for bnj852 30090. 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 30090* Technical lemma for bnj69 30177. 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 30091* Technical lemma for bnj69 30177. 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 30092* Technical lemma for bnj69 30177. 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 30093* Technical lemma for bnj69 30177. 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 30094* Technical lemma for bnj69 30177. 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 30095* 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 30096 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))
 
Theorembnj893 30097 Property of trCl. Under certain conditions, the transitive closure of 𝑋 in 𝐴 by 𝑅 is a set. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
 
Theorembnj900 30098* Technical lemma for bnj69 30177. 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 ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       (𝑓𝐵 → ∅ ∈ dom 𝑓)
 
Theorembnj906 30099 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
 
Theorembnj908 30100* Technical lemma for bnj69 30177. 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⟩})       ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝐺 Fn 𝑛𝜑″𝜓″))
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