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Theorem bnj153 31257
 Description: Technical lemma for bnj852 31298. 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.)
Hypotheses
Ref Expression
bnj153.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj153.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj153.3 𝐷 = (ω ∖ {∅})
bnj153.4 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
bnj153.5 (𝜏 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜃))
Assertion
Ref Expression
bnj153 (𝑛 = 1𝑜 → ((𝑛𝐷𝜏) → 𝜃))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑥,𝑦,𝑛   𝑅,𝑓,𝑖,𝑥,𝑦,𝑛   𝑚,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜃(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑚)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝑅(𝑚)

Proof of Theorem bnj153
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 bnj153.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2 bnj153.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj153.3 . 2 𝐷 = (ω ∖ {∅})
4 bnj153.4 . 2 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
5 bnj153.5 . 2 (𝜏 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜃))
6 biid 251 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
7 biid 251 . . . 4 ([1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜑)
81, 7bnj118 31246 . . 3 ([1𝑜 / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
98bicomi 214 . 2 ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [1𝑜 / 𝑛]𝜑)
10 bnj105 31099 . . . 4 1𝑜 ∈ V
112, 10bnj92 31239 . . 3 ([1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
1211bicomi 214 . 2 (∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [1𝑜 / 𝑛]𝜓)
13 biid 251 . 2 ([1𝑜 / 𝑛]𝜃[1𝑜 / 𝑛]𝜃)
14 biid 251 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
15 biid 251 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
16 biid 251 . . . . 5 ([1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ [1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
17 biid 251 . . . . 5 ([1𝑜 / 𝑛]𝜓[1𝑜 / 𝑛]𝜓)
186, 16, 7, 17bnj121 31247 . . . 4 ([1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓)))
198anbi2i 732 . . . . . . 7 ((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑) ↔ (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)))
2019, 11anbi12i 735 . . . . . 6 (((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑) ∧ [1𝑜 / 𝑛]𝜓) ↔ ((𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
21 df-3an 1074 . . . . . 6 ((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ ((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑) ∧ [1𝑜 / 𝑛]𝜓))
22 df-3an 1074 . . . . . 6 ((𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ((𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
2320, 21, 223bitr4i 292 . . . . 5 ((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
2423imbi2i 325 . . . 4 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
2518, 24bitri 264 . . 3 ([1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
2625bicomi 214 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ [1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
27 eqid 2760 . 2 {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
28 biid 251 . 2 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
29 biid 251 . 2 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3026sbcbii 3632 . . 3 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
31 biid 251 . . . . 5 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑)
32 biid 251 . . . . 5 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓)
33 biid 251 . . . . 5 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
3427, 31, 32, 33, 18bnj124 31248 . . . 4 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓)))
351, 7, 31, 27bnj125 31249 . . . . . . . 8 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑 ↔ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅))
3635anbi2i 732 . . . . . . 7 (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑) ↔ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅)))
372, 17, 32, 27bnj126 31250 . . . . . . 7 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3836, 37anbi12i 735 . . . . . 6 ((({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑) ∧ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓) ↔ (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅))))
39 df-3an 1074 . . . . . 6 (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓) ↔ (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑) ∧ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓))
40 df-3an 1074 . . . . . 6 (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅))))
4138, 39, 403bitr4i 292 . . . . 5 (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓) ↔ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅))))
4241imbi2i 325 . . . 4 (((𝑅 FrSe 𝐴𝑥𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
4334, 42bitri 264 . . 3 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
4430, 43bitr2i 265 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
45 biid 251 . 2 ((𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
46 biid 251 . . . . 5 ((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ (𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
47 biid 251 . . . . 5 ([𝑔 / 𝑓](𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ [𝑔 / 𝑓](𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
48 biid 251 . . . . 5 ([𝑔 / 𝑓][1𝑜 / 𝑛]𝜑[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑)
49 biid 251 . . . . 5 ([𝑔 / 𝑓][1𝑜 / 𝑛]𝜓[𝑔 / 𝑓][1𝑜 / 𝑛]𝜓)
5046, 47, 48, 49bnj156 31103 . . . 4 ([𝑔 / 𝑓](𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ (𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑[𝑔 / 𝑓][1𝑜 / 𝑛]𝜓))
5148, 8bnj154 31255 . . . . . . 7 ([𝑔 / 𝑓][1𝑜 / 𝑛]𝜑 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
5251anbi2i 732 . . . . . 6 ((𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑) ↔ (𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
5317, 11bitri 264 . . . . . . 7 ([1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
5449, 53bnj155 31256 . . . . . 6 ([𝑔 / 𝑓][1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
5552, 54anbi12i 735 . . . . 5 (((𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑) ∧ [𝑔 / 𝑓][1𝑜 / 𝑛]𝜓) ↔ ((𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))))
56 df-3an 1074 . . . . 5 ((𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑[𝑔 / 𝑓][1𝑜 / 𝑛]𝜓) ↔ ((𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑) ∧ [𝑔 / 𝑓][1𝑜 / 𝑛]𝜓))
57 df-3an 1074 . . . . 5 ((𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ((𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))))
5855, 56, 573bitr4i 292 . . . 4 ((𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑[𝑔 / 𝑓][1𝑜 / 𝑛]𝜓) ↔ (𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))))
5950, 58bitri 264 . . 3 ([𝑔 / 𝑓](𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ (𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))))
6023sbcbii 3632 . . 3 ([𝑔 / 𝑓](𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ [𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
6159, 60bitr3i 266 . 2 ((𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ [𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
62 biid 251 . 2 ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
63 biid 251 . 2 ([𝑔 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [𝑔 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
641, 2, 3, 4, 5, 6, 9, 12, 13, 14, 15, 26, 27, 28, 29, 44, 45, 61, 62, 63bnj151 31254 1 (𝑛 = 1𝑜 → ((𝑛𝐷𝜏) → 𝜃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632  ∃wex 1853   ∈ wcel 2139  ∃!weu 2607  ∃*wmo 2608  ∀wral 3050  [wsbc 3576   ∖ cdif 3712  ∅c0 4058  {csn 4321  ⟨cop 4327  ∪ ciun 4672   class class class wbr 4804   E cep 5178  suc csuc 5886   Fn wfn 6044  ‘cfv 6049  ωcom 7230  1𝑜c1o 7722   predc-bnj14 31063   FrSe w-bnj15 31067 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-1o 7729  df-bnj13 31066  df-bnj15 31068 This theorem is referenced by:  bnj852  31298
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