Theorem axdc4uzlem 14153

Description: Lemma for axdc4uz 14154. (Contributed by Mario Carneiro, 8-Jan-2014.) (Revised by Mario Carneiro, 26-Dec-2014.)

Hypotheses

Ref	Expression
axdc4uz.1	⊢ 𝑀 ∈ ℤ
axdc4uz.2	⊢ 𝑍 = (ℤ≥‘𝑀)
axdc4uz.3	⊢ 𝐴 ∈ V
axdc4uz.4	⊢ 𝐺 = (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω)
axdc4uz.5	⊢ 𝐻 = (𝑛 ∈ ω, 𝑥 ∈ 𝐴 ↦ ((𝐺‘𝑛)𝐹𝑥))

Assertion

Ref	Expression
axdc4uzlem	⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))

Distinct variable groups:   𝑔,𝑘,𝑛,𝑥,𝐴   𝐶,𝑔   𝑔,𝐹,𝑘,𝑛,𝑥   𝑦,𝑔,𝑀,𝑘,𝑛,𝑥   𝑔,𝑍,𝑛,𝑥   𝑔,𝐺,𝑘,𝑛,𝑥   𝑘,𝐻

Allowed substitution hints:   𝐴(𝑦)   𝐶(𝑥, 𝑦, 𝑘, 𝑛)   𝐹(𝑦)   𝐺(𝑦)   𝐻(𝑥, 𝑦, 𝑔, 𝑛)   𝑍(𝑦, 𝑘)

Proof of Theorem axdc4uzlem

Dummy variables 𝑓 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axdc4uz.1	. . . . . . . . . . 11 ⊢ 𝑀 ∈ ℤ
2		axdc4uz.4	. . . . . . . . . . 11 ⊢ 𝐺 = (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω)
3	1, 2	om2uzf1oi 14123	. . . . . . . . . 10 ⊢ 𝐺:ω⟶1-1onto(ℤ≥‘𝑀)
4		axdc4uz.2	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ≥‘𝑀)
5		f1oeq3 6836	. . . . . . . . . . 11 ⊢ (𝑍 = (ℤ≥‘𝑀) → (𝐺:ω⟶1-1onto𝑍 ↔ 𝐺:ω⟶1-1onto(ℤ≥‘𝑀)))
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐺:ω⟶1-1onto𝑍 ↔ 𝐺:ω⟶1-1onto(ℤ≥‘𝑀))
7	3, 6	mpbir 234	. . . . . . . . 9 ⊢ 𝐺:ω⟶1-1onto𝑍
8		f1of 6846	. . . . . . . . 9 ⊢ (𝐺:ω⟶1-1onto𝑍 → 𝐺:ω⟶𝑍)
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ 𝐺:ω⟶𝑍
10	9	ffvelcdmi 7107	. . . . . . 7 ⊢ (𝑛 ∈ ω → (𝐺‘𝑛) ∈ 𝑍)
11		fovcdm 7615	. . . . . . 7 ⊢ ((𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) ∧ (𝐺‘𝑛) ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
12	10, 11	syl3an2 1182	. . . . . 6 ⊢ ((𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) ∧ 𝑛 ∈ ω ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
13	12	3expb 1138	. . . . 5 ⊢ ((𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ 𝐴)) → ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
14	13	ralrimivva 3209	. . . 4 ⊢ (𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) → ∀𝑛 ∈ ω ∀𝑥 ∈ 𝐴 ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
15		axdc4uz.5	. . . . 5 ⊢ 𝐻 = (𝑛 ∈ ω, 𝑥 ∈ 𝐴 ↦ ((𝐺‘𝑛)𝐹𝑥))
16	15	fmpo 8104	. . . 4 ⊢ (∀𝑛 ∈ ω ∀𝑥 ∈ 𝐴 ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}) ↔ 𝐻:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅}))
17	14, 16	sylib 221	. . 3 ⊢ (𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) → 𝐻:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅}))
18		axdc4uz.3	. . . 4 ⊢ 𝐴 ∈ V
19	18	axdc4 10561	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐻:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑓(𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))))
20	17, 19	sylan2 605	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑓(𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))))
21		f1ocnv 6859	. . . . . . 7 ⊢ (𝐺:ω⟶1-1onto𝑍 → ◡𝐺:𝑍⟶1-1ontoω)
22		f1of 6846	. . . . . . 7 ⊢ (◡𝐺:𝑍⟶1-1ontoω → ◡𝐺:𝑍⟶ω)
23	7, 21, 22	mp2b 10	. . . . . 6 ⊢ ◡𝐺:𝑍⟶ω
24		fco 6756	. . . . . 6 ⊢ ((𝑓:ω⟶𝐴 ∧ ◡𝐺:𝑍⟶ω) → (𝑓 ∘ ◡𝐺):𝑍⟶𝐴)
25	23, 24	mpan2 704	. . . . 5 ⊢ (𝑓:ω⟶𝐴 → (𝑓 ∘ ◡𝐺):𝑍⟶𝐴)
26	25	3ad2ant1 1151	. . . 4 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → (𝑓 ∘ ◡𝐺):𝑍⟶𝐴)
27		uzid 13007	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
28	1, 27	ax-mp 5	. . . . . . . 8 ⊢ 𝑀 ∈ (ℤ≥‘𝑀)
29	28, 4	eleqtrri 2863	. . . . . . 7 ⊢ 𝑀 ∈ 𝑍
30		fvco3 7008	. . . . . . 7 ⊢ ((◡𝐺:𝑍⟶ω ∧ 𝑀 ∈ 𝑍) → ((𝑓 ∘ ◡𝐺)‘𝑀) = (𝑓‘(◡𝐺‘𝑀)))
31	23, 29, 30	mp2an 705	. . . . . 6 ⊢ ((𝑓 ∘ ◡𝐺)‘𝑀) = (𝑓‘(◡𝐺‘𝑀))
32	1, 2	om2uz0i 14117	. . . . . . . 8 ⊢ (𝐺‘∅) = 𝑀
33		peano1 7924	. . . . . . . . 9 ⊢ ∅ ∈ ω
34		f1ocnvfv 7309	. . . . . . . . 9 ⊢ ((𝐺:ω⟶1-1onto𝑍 ∧ ∅ ∈ ω) → ((𝐺‘∅) = 𝑀 → (◡𝐺‘𝑀) = ∅))
35	7, 33, 34	mp2an 705	. . . . . . . 8 ⊢ ((𝐺‘∅) = 𝑀 → (◡𝐺‘𝑀) = ∅)
36	32, 35	ax-mp 5	. . . . . . 7 ⊢ (◡𝐺‘𝑀) = ∅
37	36	fveq2i 6911	. . . . . 6 ⊢ (𝑓‘(◡𝐺‘𝑀)) = (𝑓‘∅)
38	31, 37	eqtri 2786	. . . . 5 ⊢ ((𝑓 ∘ ◡𝐺)‘𝑀) = (𝑓‘∅)
39		simp2 1155	. . . . 5 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → (𝑓‘∅) = 𝐶)
40	38, 39	eqtrid 2810	. . . 4 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → ((𝑓 ∘ ◡𝐺)‘𝑀) = 𝐶)
41	23	ffvelcdmi 7107	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 → (◡𝐺‘𝑘) ∈ ω)
42	41	adantl 487	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (◡𝐺‘𝑘) ∈ ω)
43		suceq 6449	. . . . . . . . . . . 12 ⊢ (𝑚 = (◡𝐺‘𝑘) → suc 𝑚 = suc (◡𝐺‘𝑘))
44	43	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑚 = (◡𝐺‘𝑘) → (𝑓‘suc 𝑚) = (𝑓‘suc (◡𝐺‘𝑘)))
45		id 23	. . . . . . . . . . . 12 ⊢ (𝑚 = (◡𝐺‘𝑘) → 𝑚 = (◡𝐺‘𝑘))
46		fveq2 6908	. . . . . . . . . . . 12 ⊢ (𝑚 = (◡𝐺‘𝑘) → (𝑓‘𝑚) = (𝑓‘(◡𝐺‘𝑘)))
47	45, 46	oveq12d 7461	. . . . . . . . . . 11 ⊢ (𝑚 = (◡𝐺‘𝑘) → (𝑚𝐻(𝑓‘𝑚)) = ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))))
48	44, 47	eleq12d 2858	. . . . . . . . . 10 ⊢ (𝑚 = (◡𝐺‘𝑘) → ((𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚)) ↔ (𝑓‘suc (◡𝐺‘𝑘)) ∈ ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘)))))
49	48	rspcv 3576	. . . . . . . . 9 ⊢ ((◡𝐺‘𝑘) ∈ ω → (∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚)) → (𝑓‘suc (◡𝐺‘𝑘)) ∈ ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘)))))
50	42, 49	syl 18	. . . . . . . 8 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚)) → (𝑓‘suc (◡𝐺‘𝑘)) ∈ ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘)))))
51	4	peano2uzs 13056	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (𝑘 + 1) ∈ 𝑍)
52		fvco3 7008	. . . . . . . . . . . 12 ⊢ ((◡𝐺:𝑍⟶ω ∧ (𝑘 + 1) ∈ 𝑍) → ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) = (𝑓‘(◡𝐺‘(𝑘 + 1))))
53	23, 51, 52	sylancr 599	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) = (𝑓‘(◡𝐺‘(𝑘 + 1))))
54	1, 2	om2uzsuci 14118	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐺‘𝑘) ∈ ω → (𝐺‘suc (◡𝐺‘𝑘)) = ((𝐺‘(◡𝐺‘𝑘)) + 1))
55	41, 54	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘suc (◡𝐺‘𝑘)) = ((𝐺‘(◡𝐺‘𝑘)) + 1))
56		f1ocnvfv2 7308	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:ω⟶1-1onto𝑍 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝑘)) = 𝑘)
57	7, 56	mpan 703	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘(◡𝐺‘𝑘)) = 𝑘)
58	57	oveq1d 7458	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 → ((𝐺‘(◡𝐺‘𝑘)) + 1) = (𝑘 + 1))
59	55, 58	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘suc (◡𝐺‘𝑘)) = (𝑘 + 1))
60		peano2 7925	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐺‘𝑘) ∈ ω → suc (◡𝐺‘𝑘) ∈ ω)
61	41, 60	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 → suc (◡𝐺‘𝑘) ∈ ω)
62		f1ocnvfv 7309	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ω⟶1-1onto𝑍 ∧ suc (◡𝐺‘𝑘) ∈ ω) → ((𝐺‘suc (◡𝐺‘𝑘)) = (𝑘 + 1) → (◡𝐺‘(𝑘 + 1)) = suc (◡𝐺‘𝑘)))
63	7, 61, 62	sylancr 599	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → ((𝐺‘suc (◡𝐺‘𝑘)) = (𝑘 + 1) → (◡𝐺‘(𝑘 + 1)) = suc (◡𝐺‘𝑘)))
64	59, 63	mpd 16	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (◡𝐺‘(𝑘 + 1)) = suc (◡𝐺‘𝑘))
65	64	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → (𝑓‘(◡𝐺‘(𝑘 + 1))) = (𝑓‘suc (◡𝐺‘𝑘)))
66	53, 65	eqtr2d 2799	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 → (𝑓‘suc (◡𝐺‘𝑘)) = ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)))
67	66	adantl 487	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (𝑓‘suc (◡𝐺‘𝑘)) = ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)))
68		ffvelcdm 7105	. . . . . . . . . . . 12 ⊢ ((𝑓:ω⟶𝐴 ∧ (◡𝐺‘𝑘) ∈ ω) → (𝑓‘(◡𝐺‘𝑘)) ∈ 𝐴)
69	41, 68	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (𝑓‘(◡𝐺‘𝑘)) ∈ 𝐴)
70		fveq2 6908	. . . . . . . . . . . . 13 ⊢ (𝑛 = (◡𝐺‘𝑘) → (𝐺‘𝑛) = (𝐺‘(◡𝐺‘𝑘)))
71	70	oveq1d 7458	. . . . . . . . . . . 12 ⊢ (𝑛 = (◡𝐺‘𝑘) → ((𝐺‘𝑛)𝐹𝑥) = ((𝐺‘(◡𝐺‘𝑘))𝐹𝑥))
72		oveq2 7451	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘(◡𝐺‘𝑘)) → ((𝐺‘(◡𝐺‘𝑘))𝐹𝑥) = ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))))
73		ovex 7476	. . . . . . . . . . . 12 ⊢ ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))) ∈ V
74	71, 72, 15, 73	ovmpo 7604	. . . . . . . . . . 11 ⊢ (((◡𝐺‘𝑘) ∈ ω ∧ (𝑓‘(◡𝐺‘𝑘)) ∈ 𝐴) → ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))) = ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))))
75	42, 69, 74	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))) = ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))))
76		fvco3 7008	. . . . . . . . . . . . . 14 ⊢ ((◡𝐺:𝑍⟶ω ∧ 𝑘 ∈ 𝑍) → ((𝑓 ∘ ◡𝐺)‘𝑘) = (𝑓‘(◡𝐺‘𝑘)))
77	23, 76	mpan 703	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → ((𝑓 ∘ ◡𝐺)‘𝑘) = (𝑓‘(◡𝐺‘𝑘)))
78	77	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (𝑓‘(◡𝐺‘𝑘)) = ((𝑓 ∘ ◡𝐺)‘𝑘))
79	57, 78	oveq12d 7461	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
80	79	adantl 487	. . . . . . . . . 10 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
81	75, 80	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
82	67, 81	eleq12d 2858	. . . . . . . 8 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((𝑓‘suc (◡𝐺‘𝑘)) ∈ ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))) ↔ ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘))))
83	50, 82	sylibd 242	. . . . . . 7 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚)) → ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘))))
84	83	impancom 457	. . . . . 6 ⊢ ((𝑓:ω⟶𝐴 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → (𝑘 ∈ 𝑍 → ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘))))
85	84	ralrimiv 3157	. . . . 5 ⊢ ((𝑓:ω⟶𝐴 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → ∀𝑘 ∈ 𝑍 ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
86	85	3adant2 1149	. . . 4 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → ∀𝑘 ∈ 𝑍 ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
87		vex 3458	. . . . . 6 ⊢ 𝑓 ∈ V
88		rdgfun 8452	. . . . . . . . 9 ⊢ Fun rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀)
89		omex 9671	. . . . . . . . 9 ⊢ ω ∈ V
90		resfunexg 7244	. . . . . . . . 9 ⊢ ((Fun rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω) ∈ V)
91	88, 89, 90	mp2an 705	. . . . . . . 8 ⊢ (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω) ∈ V
92	2, 91	eqeltri 2860	. . . . . . 7 ⊢ 𝐺 ∈ V
93	92	cnvex 7961	. . . . . 6 ⊢ ◡𝐺 ∈ V
94	87, 93	coex 7966	. . . . 5 ⊢ (𝑓 ∘ ◡𝐺) ∈ V
95		feq1 6709	. . . . . 6 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔:𝑍⟶𝐴 ↔ (𝑓 ∘ ◡𝐺):𝑍⟶𝐴))
96		fveq1 6907	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔‘𝑀) = ((𝑓 ∘ ◡𝐺)‘𝑀))
97	96	eqeq1d 2765	. . . . . 6 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → ((𝑔‘𝑀) = 𝐶 ↔ ((𝑓 ∘ ◡𝐺)‘𝑀) = 𝐶))
98		fveq1 6907	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔‘(𝑘 + 1)) = ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)))
99		fveq1 6907	. . . . . . . . 9 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔‘𝑘) = ((𝑓 ∘ ◡𝐺)‘𝑘))
100	99	oveq2d 7459	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑘𝐹(𝑔‘𝑘)) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
101	98, 100	eleq12d 2858	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘))))
102	101	ralbidv 3189	. . . . . 6 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ ∀𝑘 ∈ 𝑍 ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘))))
103	95, 97, 102	3anbi123d 1465	. . . . 5 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → ((𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))) ↔ ((𝑓 ∘ ◡𝐺):𝑍⟶𝐴 ∧ ((𝑓 ∘ ◡𝐺)‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))))
104	94, 103	spcev 3564	. . . 4 ⊢ (((𝑓 ∘ ◡𝐺):𝑍⟶𝐴 ∧ ((𝑓 ∘ ◡𝐺)‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘))) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))
105	26, 40, 86, 104	syl3anc 1399	. . 3 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))
106	105	exlimiv 1965	. 2 ⊢ (∃𝑓(𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))
107	20, 106	syl 18	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ∖ cdif 4048  ∅c0 4281  𝒫 cpw 4560  {csn 4588   ↦ cmpt 5189   × cxp 5653  ^◡ccnv 5654   ↾ cres 5657   ∘ ccom 5659  suc csuc 6382  Fun wfun 6552  ⟶wf 6554  ⟶1-1ontowf1o 6557  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  ωcom 7901  reccrdg 8445  1c1 11228   + caddc 11230  ℤcz 12720  ℤ_≥cuz 12992

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669  ax-dc 10551  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  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 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-n0 12634  df-z 12721  df-uz 12993

This theorem is used by:  axdc4uz  14154