Theorem bnj153 32760

Description: Technical lemma for bnj852 32801. 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 260	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
7		biid 260	. . . 4 ⊢ ([1o / 𝑛]𝜑 ↔ [1o / 𝑛]𝜑)
8	1, 7	bnj118 32749	. . 3 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
9	8	bicomi 223	. 2 ⊢ ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [1o / 𝑛]𝜑)
10		bnj105 32603	. . . 4 ⊢ 1o ∈ V
11	2, 10	bnj92 32742	. . 3 ⊢ ([1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
12	11	bicomi 223	. 2 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [1o / 𝑛]𝜓)
13		biid 260	. 2 ⊢ ([1o / 𝑛]𝜃 ↔ [1o / 𝑛]𝜃)
14		biid 260	. 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 260	. 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 260	. . . . 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 260	. . . . 5 ⊢ ([1o / 𝑛]𝜓 ↔ [1o / 𝑛]𝜓)
18	6, 16, 7, 17	bnj121 32750	. . . 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 626	. . . . . 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 1087	. . . . . 6 ⊢ ((𝑓 Fn 1o ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓) ↔ ((𝑓 Fn 1o ∧ [1o / 𝑛]𝜑) ∧ [1o / 𝑛]𝜓))
22		df-3an 1087	. . . . . 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 302	. . . . 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 335	. . . 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 274	. . 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 223	. 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 2738	. 2 ⊢ {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
28		biid 260	. 2 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
29		biid 260	. 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 3772	. . 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 260	. . . . 5 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜑 ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜑)
32		biid 260	. . . . 5 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜓 ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜓)
33		biid 260	. . . . 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 32751	. . . 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 32752	. . . . . . . 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 32753	. . . . . . 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 626	. . . . . 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 1087	. . . . . 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 1087	. . . . . 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 302	. . . . 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 335	. . . 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 274	. . 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 275	. 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 260	. 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 260	. . . . 5 ⊢ ((𝑓 Fn 1o ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓) ↔ (𝑓 Fn 1o ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓))
47		biid 260	. . . . 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 260	. . . . 5 ⊢ ([𝑔 / 𝑓][1o / 𝑛]𝜑 ↔ [𝑔 / 𝑓][1o / 𝑛]𝜑)
49		biid 260	. . . . 5 ⊢ ([𝑔 / 𝑓][1o / 𝑛]𝜓 ↔ [𝑔 / 𝑓][1o / 𝑛]𝜓)
50	46, 47, 48, 49	bnj156 32607	. . . 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 32758	. . . . . . 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 274	. . . . . . 7 ⊢ ([1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
54	49, 53	bnj155 32759	. . . . . 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 626	. . . . 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 1087	. . . . 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 1087	. . . . 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 302	. . . 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 274	. . 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 3772	. . 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 276	. 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 260	. 2 ⊢ ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
63		biid 260	. 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 32757	1 ⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃*wmo 2538  ∃!weu 2568  ∀wral 3063  [wsbc 3711   ∖ cdif 3880  ∅c0 4253  {csn 4558  ⟨cop 4564  ∪ ciun 4921   class class class wbr 5070   E cep 5485  suc csuc 6253   Fn wfn 6413  ‘cfv 6418  ωcom 7687  1_oc1o 8260   predc-bnj14 32567   FrSe w-bnj15 32571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-1o 8267  df-bnj13 32570  df-bnj15 32572

This theorem is referenced by:  bnj852  32801