Theorem fin23lem34 10437

Description: Lemma for fin23 10480. Establish induction invariants on 𝑌 which parameterizes our contradictory chain of subsets. In this section, ℎ is the hypothetically assumed family of subsets, 𝑔 is the ground set, and 𝑖 is the induction function constructed in the previous section. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypotheses

Ref	Expression
fin23lem33.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
fin23lem.f	⊢ (𝜑 → ℎ:ω–1-1→V)
fin23lem.g	⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
fin23lem.h	⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
fin23lem.i	⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)

Assertion

Ref	Expression
fin23lem34	⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))

Distinct variable groups:   𝑔,𝑎,𝑖,𝑗,𝑥   𝐴,𝑎,𝑗   ℎ,𝑎,𝐺,𝑔,𝑖,𝑗,𝑥   𝐹,𝑎   𝜑,𝑎,𝑗   𝑌,𝑎,𝑗

Allowed substitution hints:   𝜑(𝑥, 𝑔, ℎ, 𝑖)   𝐴(𝑥, 𝑔, ℎ, 𝑖)   𝐹(𝑥, 𝑔, ℎ, 𝑖, 𝑗)   𝑌(𝑥, 𝑔, ℎ, 𝑖)

Proof of Theorem fin23lem34

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6896	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑌‘𝑎) = (𝑌‘∅))
2		f1eq1 6784	. . . . . 6 ⊢ ((𝑌‘𝑎) = (𝑌‘∅) → ((𝑌‘𝑎):ω–1-1→V ↔ (𝑌‘∅):ω–1-1→V))
3	1, 2	syl 18	. . . . 5 ⊢ (𝑎 = ∅ → ((𝑌‘𝑎):ω–1-1→V ↔ (𝑌‘∅):ω–1-1→V))
4	1	rneqd 5925	. . . . . . 7 ⊢ (𝑎 = ∅ → ran (𝑌‘𝑎) = ran (𝑌‘∅))
5	4	unieqd 4883	. . . . . 6 ⊢ (𝑎 = ∅ → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘∅))
6	5	sseq1d 3944	. . . . 5 ⊢ (𝑎 = ∅ → (∪ ran (𝑌‘𝑎) ⊆ 𝐺 ↔ ∪ ran (𝑌‘∅) ⊆ 𝐺))
7	3, 6	anbi12d 644	. . . 4 ⊢ (𝑎 = ∅ → (((𝑌‘𝑎):ω–1-1→V ∧ ∪ ran (𝑌‘𝑎) ⊆ 𝐺) ↔ ((𝑌‘∅):ω–1-1→V ∧ ∪ ran (𝑌‘∅) ⊆ 𝐺)))
8	7	imbi2d 343	. . 3 ⊢ (𝑎 = ∅ → ((𝜑 → ((𝑌‘𝑎):ω–1-1→V ∧ ∪ ran (𝑌‘𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌‘∅):ω–1-1→V ∧ ∪ ran (𝑌‘∅) ⊆ 𝐺))))
9		fveq2 6896	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑌‘𝑎) = (𝑌‘𝑏))
10		f1eq1 6784	. . . . . 6 ⊢ ((𝑌‘𝑎) = (𝑌‘𝑏) → ((𝑌‘𝑎):ω–1-1→V ↔ (𝑌‘𝑏):ω–1-1→V))
11	9, 10	syl 18	. . . . 5 ⊢ (𝑎 = 𝑏 → ((𝑌‘𝑎):ω–1-1→V ↔ (𝑌‘𝑏):ω–1-1→V))
12	9	rneqd 5925	. . . . . . 7 ⊢ (𝑎 = 𝑏 → ran (𝑌‘𝑎) = ran (𝑌‘𝑏))
13	12	unieqd 4883	. . . . . 6 ⊢ (𝑎 = 𝑏 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝑏))
14	13	sseq1d 3944	. . . . 5 ⊢ (𝑎 = 𝑏 → (∪ ran (𝑌‘𝑎) ⊆ 𝐺 ↔ ∪ ran (𝑌‘𝑏) ⊆ 𝐺))
15	11, 14	anbi12d 644	. . . 4 ⊢ (𝑎 = 𝑏 → (((𝑌‘𝑎):ω–1-1→V ∧ ∪ ran (𝑌‘𝑎) ⊆ 𝐺) ↔ ((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺)))
16	15	imbi2d 343	. . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → ((𝑌‘𝑎):ω–1-1→V ∧ ∪ ran (𝑌‘𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺))))
17		fveq2 6896	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → (𝑌‘𝑎) = (𝑌‘suc 𝑏))
18		f1eq1 6784	. . . . . 6 ⊢ ((𝑌‘𝑎) = (𝑌‘suc 𝑏) → ((𝑌‘𝑎):ω–1-1→V ↔ (𝑌‘suc 𝑏):ω–1-1→V))
19	17, 18	syl 18	. . . . 5 ⊢ (𝑎 = suc 𝑏 → ((𝑌‘𝑎):ω–1-1→V ↔ (𝑌‘suc 𝑏):ω–1-1→V))
20	17	rneqd 5925	. . . . . . 7 ⊢ (𝑎 = suc 𝑏 → ran (𝑌‘𝑎) = ran (𝑌‘suc 𝑏))
21	20	unieqd 4883	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘suc 𝑏))
22	21	sseq1d 3944	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (∪ ran (𝑌‘𝑎) ⊆ 𝐺 ↔ ∪ ran (𝑌‘suc 𝑏) ⊆ 𝐺))
23	19, 22	anbi12d 644	. . . 4 ⊢ (𝑎 = suc 𝑏 → (((𝑌‘𝑎):ω–1-1→V ∧ ∪ ran (𝑌‘𝑎) ⊆ 𝐺) ↔ ((𝑌‘suc 𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘suc 𝑏) ⊆ 𝐺)))
24	23	imbi2d 343	. . 3 ⊢ (𝑎 = suc 𝑏 → ((𝜑 → ((𝑌‘𝑎):ω–1-1→V ∧ ∪ ran (𝑌‘𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌‘suc 𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘suc 𝑏) ⊆ 𝐺))))
25		fveq2 6896	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑌‘𝑎) = (𝑌‘𝐴))
26		f1eq1 6784	. . . . . 6 ⊢ ((𝑌‘𝑎) = (𝑌‘𝐴) → ((𝑌‘𝑎):ω–1-1→V ↔ (𝑌‘𝐴):ω–1-1→V))
27	25, 26	syl 18	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑌‘𝑎):ω–1-1→V ↔ (𝑌‘𝐴):ω–1-1→V))
28	25	rneqd 5925	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ran (𝑌‘𝑎) = ran (𝑌‘𝐴))
29	28	unieqd 4883	. . . . . 6 ⊢ (𝑎 = 𝐴 → ∪ ran (𝑌‘𝑎) = ∪ ran (𝑌‘𝐴))
30	29	sseq1d 3944	. . . . 5 ⊢ (𝑎 = 𝐴 → (∪ ran (𝑌‘𝑎) ⊆ 𝐺 ↔ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
31	27, 30	anbi12d 644	. . . 4 ⊢ (𝑎 = 𝐴 → (((𝑌‘𝑎):ω–1-1→V ∧ ∪ ran (𝑌‘𝑎) ⊆ 𝐺) ↔ ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺)))
32	31	imbi2d 343	. . 3 ⊢ (𝑎 = 𝐴 → ((𝜑 → ((𝑌‘𝑎):ω–1-1→V ∧ ∪ ran (𝑌‘𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))))
33		fin23lem.f	. . . 4 ⊢ (𝜑 → ℎ:ω–1-1→V)
34		fin23lem.g	. . . 4 ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
35		fin23lem.i	. . . . . . . 8 ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)
36	35	fveq1i 6897	. . . . . . 7 ⊢ (𝑌‘∅) = ((rec(𝑖, ℎ) ↾ ω)‘∅)
37		fr0g 8458	. . . . . . . 8 ⊢ (ℎ ∈ V → ((rec(𝑖, ℎ) ↾ ω)‘∅) = ℎ)
38	37	elv 3459	. . . . . . 7 ⊢ ((rec(𝑖, ℎ) ↾ ω)‘∅) = ℎ
39	36, 38	eqtri 2786	. . . . . 6 ⊢ (𝑌‘∅) = ℎ
40		f1eq1 6784	. . . . . 6 ⊢ ((𝑌‘∅) = ℎ → ((𝑌‘∅):ω–1-1→V ↔ ℎ:ω–1-1→V))
41	39, 40	ax-mp 5	. . . . 5 ⊢ ((𝑌‘∅):ω–1-1→V ↔ ℎ:ω–1-1→V)
42	39	rneqi 5924	. . . . . . 7 ⊢ ran (𝑌‘∅) = ran ℎ
43	42	unieqi 4882	. . . . . 6 ⊢ ∪ ran (𝑌‘∅) = ∪ ran ℎ
44	43	sseq1i 3941	. . . . 5 ⊢ (∪ ran (𝑌‘∅) ⊆ 𝐺 ↔ ∪ ran ℎ ⊆ 𝐺)
45	41, 44	anbi12i 640	. . . 4 ⊢ (((𝑌‘∅):ω–1-1→V ∧ ∪ ran (𝑌‘∅) ⊆ 𝐺) ↔ (ℎ:ω–1-1→V ∧ ∪ ran ℎ ⊆ 𝐺))
46	33, 34, 45	sylanbrc 595	. . 3 ⊢ (𝜑 → ((𝑌‘∅):ω–1-1→V ∧ ∪ ran (𝑌‘∅) ⊆ 𝐺))
47		fin23lem.h	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
48		fvex 6909	. . . . . . . . . . 11 ⊢ (𝑌‘𝑏) ∈ V
49		f1eq1 6784	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑌‘𝑏) → (𝑗:ω–1-1→V ↔ (𝑌‘𝑏):ω–1-1→V))
50		rneq 5923	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑌‘𝑏) → ran 𝑗 = ran (𝑌‘𝑏))
51	50	unieqd 4883	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑌‘𝑏) → ∪ ran 𝑗 = ∪ ran (𝑌‘𝑏))
52	51	sseq1d 3944	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑌‘𝑏) → (∪ ran 𝑗 ⊆ 𝐺 ↔ ∪ ran (𝑌‘𝑏) ⊆ 𝐺))
53	49, 52	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑌‘𝑏) → ((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) ↔ ((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺)))
54		fveq2 6896	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑌‘𝑏) → (𝑖‘𝑗) = (𝑖‘(𝑌‘𝑏)))
55		f1eq1 6784	. . . . . . . . . . . . . 14 ⊢ ((𝑖‘𝑗) = (𝑖‘(𝑌‘𝑏)) → ((𝑖‘𝑗):ω–1-1→V ↔ (𝑖‘(𝑌‘𝑏)):ω–1-1→V))
56	54, 55	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑌‘𝑏) → ((𝑖‘𝑗):ω–1-1→V ↔ (𝑖‘(𝑌‘𝑏)):ω–1-1→V))
57	54	rneqd 5925	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑌‘𝑏) → ran (𝑖‘𝑗) = ran (𝑖‘(𝑌‘𝑏)))
58	57	unieqd 4883	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑌‘𝑏) → ∪ ran (𝑖‘𝑗) = ∪ ran (𝑖‘(𝑌‘𝑏)))
59	58, 51	psseq12d 4027	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑌‘𝑏) → (∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗 ↔ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏)))
60	56, 59	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑌‘𝑏) → (((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗) ↔ ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏))))
61	53, 60	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝑏) → (((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)) ↔ (((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏)))))
62	48, 61	spcv 3563	. . . . . . . . . 10 ⊢ (∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)) → (((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏))))
63	47, 62	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏))))
64	63	imp 412	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺)) → ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏)))
65		pssss 4028	. . . . . . . . . . . 12 ⊢ (∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏) → ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ ∪ ran (𝑌‘𝑏))
66		sstr 3921	. . . . . . . . . . . 12 ⊢ ((∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ ∪ ran (𝑌‘𝑏) ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺) → ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺)
67	65, 66	sylan 592	. . . . . . . . . . 11 ⊢ ((∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏) ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺) → ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺)
68	67	expcom 419	. . . . . . . . . 10 ⊢ (∪ ran (𝑌‘𝑏) ⊆ 𝐺 → (∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏) → ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺))
69	68	anim2d 624	. . . . . . . . 9 ⊢ (∪ ran (𝑌‘𝑏) ⊆ 𝐺 → (((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏)) → ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺)))
70	69	ad2antll 742	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺)) → (((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑏)) → ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺)))
71	64, 70	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺)) → ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺))
72	71	3adant1 1148	. . . . . 6 ⊢ ((𝑏 ∈ ω ∧ 𝜑 ∧ ((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺)) → ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺))
73		frsuc 8459	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ((rec(𝑖, ℎ) ↾ ω)‘suc 𝑏) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝑏)))
74	35	fveq1i 6897	. . . . . . . . 9 ⊢ (𝑌‘suc 𝑏) = ((rec(𝑖, ℎ) ↾ ω)‘suc 𝑏)
75	35	fveq1i 6897	. . . . . . . . . 10 ⊢ (𝑌‘𝑏) = ((rec(𝑖, ℎ) ↾ ω)‘𝑏)
76	75	fveq2i 6899	. . . . . . . . 9 ⊢ (𝑖‘(𝑌‘𝑏)) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝑏))
77	73, 74, 76	3eqtr4g 2823	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝑌‘suc 𝑏) = (𝑖‘(𝑌‘𝑏)))
78		f1eq1 6784	. . . . . . . . 9 ⊢ ((𝑌‘suc 𝑏) = (𝑖‘(𝑌‘𝑏)) → ((𝑌‘suc 𝑏):ω–1-1→V ↔ (𝑖‘(𝑌‘𝑏)):ω–1-1→V))
79		rneq 5923	. . . . . . . . . . 11 ⊢ ((𝑌‘suc 𝑏) = (𝑖‘(𝑌‘𝑏)) → ran (𝑌‘suc 𝑏) = ran (𝑖‘(𝑌‘𝑏)))
80	79	unieqd 4883	. . . . . . . . . 10 ⊢ ((𝑌‘suc 𝑏) = (𝑖‘(𝑌‘𝑏)) → ∪ ran (𝑌‘suc 𝑏) = ∪ ran (𝑖‘(𝑌‘𝑏)))
81	80	sseq1d 3944	. . . . . . . . 9 ⊢ ((𝑌‘suc 𝑏) = (𝑖‘(𝑌‘𝑏)) → (∪ ran (𝑌‘suc 𝑏) ⊆ 𝐺 ↔ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺))
82	78, 81	anbi12d 644	. . . . . . . 8 ⊢ ((𝑌‘suc 𝑏) = (𝑖‘(𝑌‘𝑏)) → (((𝑌‘suc 𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘suc 𝑏) ⊆ 𝐺) ↔ ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺)))
83	77, 82	syl 18	. . . . . . 7 ⊢ (𝑏 ∈ ω → (((𝑌‘suc 𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘suc 𝑏) ⊆ 𝐺) ↔ ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺)))
84	83	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑏 ∈ ω ∧ 𝜑 ∧ ((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺)) → (((𝑌‘suc 𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘suc 𝑏) ⊆ 𝐺) ↔ ((𝑖‘(𝑌‘𝑏)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝑏)) ⊆ 𝐺)))
85	72, 84	mpbird 260	. . . . 5 ⊢ ((𝑏 ∈ ω ∧ 𝜑 ∧ ((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺)) → ((𝑌‘suc 𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘suc 𝑏) ⊆ 𝐺))
86	85	3exp 1137	. . . 4 ⊢ (𝑏 ∈ ω → (𝜑 → (((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺) → ((𝑌‘suc 𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘suc 𝑏) ⊆ 𝐺))))
87	86	a2d 30	. . 3 ⊢ (𝑏 ∈ ω → ((𝜑 → ((𝑌‘𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘𝑏) ⊆ 𝐺)) → (𝜑 → ((𝑌‘suc 𝑏):ω–1-1→V ∧ ∪ ran (𝑌‘suc 𝑏) ⊆ 𝐺))))
88	8, 16, 24, 32, 46, 87	finds 7918	. 2 ⊢ (𝐴 ∈ ω → (𝜑 → ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺)))
89	88	impcom 413	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2146  {cab 2741  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ⊊ wpss 4016  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910  ran crn 5656   ↾ cres 5657  suc csuc 6371  –1-1→wf1 6544  ‘cfv 6547  (class class class)co 7430  ωcom 7887  reccrdg 8431   ↑_m cmap 8861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432

This theorem is used by:  fin23lem35  10438  fin23lem39  10441