Theorem bnj153 34856

Description: Technical lemma for bnj852 34897. 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 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj153.3	⊢ 𝐷 = (ω ∖ {∅})
bnj153.4	⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj153.5	⊢ (𝜏 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜃))

Assertion

Ref	Expression
bnj153	⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 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.

Step	Hyp	Ref	Expression
1		bnj153.1	. 2 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2		bnj153.2	. 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj153.3	. 2 ⊢ 𝐷 = (ω ∖ {∅})
4		bnj153.4	. 2 ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
5		bnj153.5	. 2 ⊢ (𝜏 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜃))
6		biid 261	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
7		biid 261	. . . 4 ⊢ ([1o / 𝑛]𝜑 ↔ [1o / 𝑛]𝜑)
8	1, 7	bnj118 34845	. . 3 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
9	8	bicomi 224	. 2 ⊢ ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [1o / 𝑛]𝜑)
10		bnj105 34700	. . . 4 ⊢ 1o ∈ V
11	2, 10	bnj92 34838	. . 3 ⊢ ([1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
12	11	bicomi 224	. 2 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [1o / 𝑛]𝜓)
13		biid 261	. 2 ⊢ ([1o / 𝑛]𝜃 ↔ [1o / 𝑛]𝜃)
14		biid 261	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
15		biid 261	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
16		biid 261	. . . . 5 ⊢ ([1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ [1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
17		biid 261	. . . . 5 ⊢ ([1o / 𝑛]𝜓 ↔ [1o / 𝑛]𝜓)
18	6, 16, 7, 17	bnj121 34846	. . . 4 ⊢ ([1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓)))
19	8	anbi2i 622	. . . . . . 7 ⊢ ((𝑓 Fn 1o ∧ [1o / 𝑛]𝜑) ↔ (𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)))
20	19, 11	anbi12i 627	. . . . . 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 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
21		df-3an 1089	. . . . . 6 ⊢ ((𝑓 Fn 1o ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓) ↔ ((𝑓 Fn 1o ∧ [1o / 𝑛]𝜑) ∧ [1o / 𝑛]𝜓))
22		df-3an 1089	. . . . . 6 ⊢ ((𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ((𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
23	20, 21, 22	3bitr4i 303	. . . . 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 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
24	23	imbi2i 336	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
25	18, 24	bitri 275	. . 3 ⊢ ([1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
26	25	bicomi 224	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ [1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
27		eqid 2740	. 2 ⊢ {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
28		biid 261	. 2 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
29		biid 261	. 2 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
30	26	sbcbii 3865	. . 3 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
31		biid 261	. . . . 5 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜑 ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜑)
32		biid 261	. . . . 5 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜓 ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜓)
33		biid 261	. . . . 5 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
34	27, 31, 32, 33, 18	bnj124 34847	. . . 4 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1o ∧ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜑 ∧ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜓)))
35	1, 7, 31, 27	bnj125 34848	. . . . . . . 8 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜑 ↔ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅))
36	35	anbi2i 622	. . . . . . 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)))
37	2, 17, 32, 27	bnj126 34849	. . . . . . 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)))
38	36, 37	anbi12i 627	. . . . . 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 1089	. . . . . 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 1089	. . . . . 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))))
41	38, 39, 40	3bitr4i 303	. . . . 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))))
42	41	imbi2i 336	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1o ∧ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜑 ∧ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ({⟨∅, 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)))))
43	34, 42	bitri 275	. . 3 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ({⟨∅, 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)))))
44	30, 43	bitr2i 276	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ({⟨∅, 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 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
45		biid 261	. 2 ⊢ ((𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
46		biid 261	. . . . 5 ⊢ ((𝑓 Fn 1o ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓) ↔ (𝑓 Fn 1o ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓))
47		biid 261	. . . . 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 261	. . . . 5 ⊢ ([𝑔 / 𝑓][1o / 𝑛]𝜑 ↔ [𝑔 / 𝑓][1o / 𝑛]𝜑)
49		biid 261	. . . . 5 ⊢ ([𝑔 / 𝑓][1o / 𝑛]𝜓 ↔ [𝑔 / 𝑓][1o / 𝑛]𝜓)
50	46, 47, 48, 49	bnj156 34704	. . . 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))
51	48, 8	bnj154 34854	. . . . . . 7 ⊢ ([𝑔 / 𝑓][1o / 𝑛]𝜑 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
52	51	anbi2i 622	. . . . . 6 ⊢ ((𝑔 Fn 1o ∧ [𝑔 / 𝑓][1o / 𝑛]𝜑) ↔ (𝑔 Fn 1o ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
53	17, 11	bitri 275	. . . . . . 7 ⊢ ([1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
54	49, 53	bnj155 34855	. . . . . 6 ⊢ ([𝑔 / 𝑓][1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))
55	52, 54	anbi12i 627	. . . . 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 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))))
56		df-3an 1089	. . . . 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 1089	. . . . 5 ⊢ ((𝑔 Fn 1o ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ((𝑔 Fn 1o ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))))
58	55, 56, 57	3bitr4i 303	. . . 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 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))))
59	50, 58	bitri 275	. . 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 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))))
60	23	sbcbii 3865	. . 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 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
61	59, 60	bitr3i 277	. 2 ⊢ ((𝑔 Fn 1o ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ [𝑔 / 𝑓](𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
62		biid 261	. 2 ⊢ ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
63		biid 261	. 2 ⊢ ([𝑔 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [𝑔 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
64	1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 15, 26, 27, 28, 29, 44, 45, 61, 62, 63	bnj151 34853	1 ⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃*wmo 2541  ∃!weu 2571  ∀wral 3067  [wsbc 3804   ∖ cdif 3973  ∅c0 4352  {csn 4648  ⟨cop 4654  ∪ ciun 5015   class class class wbr 5166   E cep 5598  suc csuc 6397   Fn wfn 6568  ‘cfv 6573  ωcom 7903  1_oc1o 8515   predc-bnj14 34664   FrSe w-bnj15 34668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-1o 8522  df-bnj13 34667  df-bnj15 34669

This theorem is referenced by:  bnj852  34897