Theorem bnj153 35015

Description: Technical lemma for bnj852 35056. 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 35004	. . 3 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
9	8	bicomi 224	. 2 ⊢ ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [1o / 𝑛]𝜑)
10		bnj105 34859	. . . 4 ⊢ 1o ∈ V
11	2, 10	bnj92 34997	. . 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 35005	. . . 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 624	. . . . . . 7 ⊢ ((𝑓 Fn 1o ∧ [1o / 𝑛]𝜑) ↔ (𝑓 Fn 1o ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)))
20	19, 11	anbi12i 629	. . . . . 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 2735	. 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 3796	. . 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 35006	. . . 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 35007	. . . . . . . 8 ⊢ ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1o / 𝑛]𝜑 ↔ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅))
36	35	anbi2i 624	. . . . . . 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 35008	. . . . . . 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 629	. . . . . 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 34863	. . . 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 35013	. . . . . . 7 ⊢ ([𝑔 / 𝑓][1o / 𝑛]𝜑 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
52	51	anbi2i 624	. . . . . 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 35014	. . . . . 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 629	. . . . 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 3796	. . 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 35012	1 ⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2536  ∃!weu 2567  ∀wral 3050  [wsbc 3739   ∖ cdif 3897  ∅c0 4284  {csn 4579  ⟨cop 4585  ∪ ciun 4945   class class class wbr 5097   E cep 5522  suc csuc 6318   Fn wfn 6486  ‘cfv 6491  ωcom 7808  1_oc1o 8390   predc-bnj14 34823   FrSe w-bnj15 34827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-1o 8397  df-bnj13 34826  df-bnj15 34828

This theorem is referenced by:  bnj852  35056