Theorem axdc4uzlem 13948

Description: Lemma for axdc4uz 13949. (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 13918	. . . . . . . . . 10 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝑀)
4		axdc4uz.2	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ≥‘𝑀)
5		f1oeq3 6790	. . . . . . . . . . 11 ⊢ (𝑍 = (ℤ≥‘𝑀) → (𝐺:ω–1-1-onto→𝑍 ↔ 𝐺:ω–1-1-onto→(ℤ≥‘𝑀)))
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐺:ω–1-1-onto→𝑍 ↔ 𝐺:ω–1-1-onto→(ℤ≥‘𝑀))
7	3, 6	mpbir 231	. . . . . . . . 9 ⊢ 𝐺:ω–1-1-onto→𝑍
8		f1of 6800	. . . . . . . . 9 ⊢ (𝐺:ω–1-1-onto→𝑍 → 𝐺:ω⟶𝑍)
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ 𝐺:ω⟶𝑍
10	9	ffvelcdmi 7055	. . . . . . 7 ⊢ (𝑛 ∈ ω → (𝐺‘𝑛) ∈ 𝑍)
11		fovcdm 7559	. . . . . . 7 ⊢ ((𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) ∧ (𝐺‘𝑛) ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
12	10, 11	syl3an2 1164	. . . . . 6 ⊢ ((𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) ∧ 𝑛 ∈ ω ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
13	12	3expb 1120	. . . . 5 ⊢ ((𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ 𝐴)) → ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
14	13	ralrimivva 3180	. . . 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 8047	. . . 4 ⊢ (∀𝑛 ∈ ω ∀𝑥 ∈ 𝐴 ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}) ↔ 𝐻:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅}))
17	14, 16	sylib 218	. . 3 ⊢ (𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) → 𝐻:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅}))
18		axdc4uz.3	. . . 4 ⊢ 𝐴 ∈ V
19	18	axdc4 10409	. . 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 593	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑓(𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))))
21		f1ocnv 6812	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→𝑍 → ◡𝐺:𝑍–1-1-onto→ω)
22		f1of 6800	. . . . . . 7 ⊢ (◡𝐺:𝑍–1-1-onto→ω → ◡𝐺:𝑍⟶ω)
23	7, 21, 22	mp2b 10	. . . . . 6 ⊢ ◡𝐺:𝑍⟶ω
24		fco 6712	. . . . . 6 ⊢ ((𝑓:ω⟶𝐴 ∧ ◡𝐺:𝑍⟶ω) → (𝑓 ∘ ◡𝐺):𝑍⟶𝐴)
25	23, 24	mpan2 691	. . . . 5 ⊢ (𝑓:ω⟶𝐴 → (𝑓 ∘ ◡𝐺):𝑍⟶𝐴)
26	25	3ad2ant1 1133	. . . 4 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → (𝑓 ∘ ◡𝐺):𝑍⟶𝐴)
27		uzid 12808	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
28	1, 27	ax-mp 5	. . . . . . . 8 ⊢ 𝑀 ∈ (ℤ≥‘𝑀)
29	28, 4	eleqtrri 2827	. . . . . . 7 ⊢ 𝑀 ∈ 𝑍
30		fvco3 6960	. . . . . . 7 ⊢ ((◡𝐺:𝑍⟶ω ∧ 𝑀 ∈ 𝑍) → ((𝑓 ∘ ◡𝐺)‘𝑀) = (𝑓‘(◡𝐺‘𝑀)))
31	23, 29, 30	mp2an 692	. . . . . 6 ⊢ ((𝑓 ∘ ◡𝐺)‘𝑀) = (𝑓‘(◡𝐺‘𝑀))
32	1, 2	om2uz0i 13912	. . . . . . . 8 ⊢ (𝐺‘∅) = 𝑀
33		peano1 7865	. . . . . . . . 9 ⊢ ∅ ∈ ω
34		f1ocnvfv 7253	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ ∅ ∈ ω) → ((𝐺‘∅) = 𝑀 → (◡𝐺‘𝑀) = ∅))
35	7, 33, 34	mp2an 692	. . . . . . . 8 ⊢ ((𝐺‘∅) = 𝑀 → (◡𝐺‘𝑀) = ∅)
36	32, 35	ax-mp 5	. . . . . . 7 ⊢ (◡𝐺‘𝑀) = ∅
37	36	fveq2i 6861	. . . . . 6 ⊢ (𝑓‘(◡𝐺‘𝑀)) = (𝑓‘∅)
38	31, 37	eqtri 2752	. . . . 5 ⊢ ((𝑓 ∘ ◡𝐺)‘𝑀) = (𝑓‘∅)
39		simp2 1137	. . . . 5 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → (𝑓‘∅) = 𝐶)
40	38, 39	eqtrid 2776	. . . 4 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → ((𝑓 ∘ ◡𝐺)‘𝑀) = 𝐶)
41	23	ffvelcdmi 7055	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 → (◡𝐺‘𝑘) ∈ ω)
42	41	adantl 481	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (◡𝐺‘𝑘) ∈ ω)
43		suceq 6400	. . . . . . . . . . . 12 ⊢ (𝑚 = (◡𝐺‘𝑘) → suc 𝑚 = suc (◡𝐺‘𝑘))
44	43	fveq2d 6862	. . . . . . . . . . 11 ⊢ (𝑚 = (◡𝐺‘𝑘) → (𝑓‘suc 𝑚) = (𝑓‘suc (◡𝐺‘𝑘)))
45		id 22	. . . . . . . . . . . 12 ⊢ (𝑚 = (◡𝐺‘𝑘) → 𝑚 = (◡𝐺‘𝑘))
46		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑚 = (◡𝐺‘𝑘) → (𝑓‘𝑚) = (𝑓‘(◡𝐺‘𝑘)))
47	45, 46	oveq12d 7405	. . . . . . . . . . 11 ⊢ (𝑚 = (◡𝐺‘𝑘) → (𝑚𝐻(𝑓‘𝑚)) = ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))))
48	44, 47	eleq12d 2822	. . . . . . . . . 10 ⊢ (𝑚 = (◡𝐺‘𝑘) → ((𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚)) ↔ (𝑓‘suc (◡𝐺‘𝑘)) ∈ ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘)))))
49	48	rspcv 3584	. . . . . . . . 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 17	. . . . . . . 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 12861	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (𝑘 + 1) ∈ 𝑍)
52		fvco3 6960	. . . . . . . . . . . 12 ⊢ ((◡𝐺:𝑍⟶ω ∧ (𝑘 + 1) ∈ 𝑍) → ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) = (𝑓‘(◡𝐺‘(𝑘 + 1))))
53	23, 51, 52	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) = (𝑓‘(◡𝐺‘(𝑘 + 1))))
54	1, 2	om2uzsuci 13913	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐺‘𝑘) ∈ ω → (𝐺‘suc (◡𝐺‘𝑘)) = ((𝐺‘(◡𝐺‘𝑘)) + 1))
55	41, 54	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘suc (◡𝐺‘𝑘)) = ((𝐺‘(◡𝐺‘𝑘)) + 1))
56		f1ocnvfv2 7252	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝑘)) = 𝑘)
57	7, 56	mpan 690	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘(◡𝐺‘𝑘)) = 𝑘)
58	57	oveq1d 7402	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 → ((𝐺‘(◡𝐺‘𝑘)) + 1) = (𝑘 + 1))
59	55, 58	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘suc (◡𝐺‘𝑘)) = (𝑘 + 1))
60		peano2 7866	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐺‘𝑘) ∈ ω → suc (◡𝐺‘𝑘) ∈ ω)
61	41, 60	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 → suc (◡𝐺‘𝑘) ∈ ω)
62		f1ocnvfv 7253	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ suc (◡𝐺‘𝑘) ∈ ω) → ((𝐺‘suc (◡𝐺‘𝑘)) = (𝑘 + 1) → (◡𝐺‘(𝑘 + 1)) = suc (◡𝐺‘𝑘)))
63	7, 61, 62	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → ((𝐺‘suc (◡𝐺‘𝑘)) = (𝑘 + 1) → (◡𝐺‘(𝑘 + 1)) = suc (◡𝐺‘𝑘)))
64	59, 63	mpd 15	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (◡𝐺‘(𝑘 + 1)) = suc (◡𝐺‘𝑘))
65	64	fveq2d 6862	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → (𝑓‘(◡𝐺‘(𝑘 + 1))) = (𝑓‘suc (◡𝐺‘𝑘)))
66	53, 65	eqtr2d 2765	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 → (𝑓‘suc (◡𝐺‘𝑘)) = ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)))
67	66	adantl 481	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (𝑓‘suc (◡𝐺‘𝑘)) = ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)))
68		ffvelcdm 7053	. . . . . . . . . . . 12 ⊢ ((𝑓:ω⟶𝐴 ∧ (◡𝐺‘𝑘) ∈ ω) → (𝑓‘(◡𝐺‘𝑘)) ∈ 𝐴)
69	41, 68	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (𝑓‘(◡𝐺‘𝑘)) ∈ 𝐴)
70		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑛 = (◡𝐺‘𝑘) → (𝐺‘𝑛) = (𝐺‘(◡𝐺‘𝑘)))
71	70	oveq1d 7402	. . . . . . . . . . . 12 ⊢ (𝑛 = (◡𝐺‘𝑘) → ((𝐺‘𝑛)𝐹𝑥) = ((𝐺‘(◡𝐺‘𝑘))𝐹𝑥))
72		oveq2 7395	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘(◡𝐺‘𝑘)) → ((𝐺‘(◡𝐺‘𝑘))𝐹𝑥) = ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))))
73		ovex 7420	. . . . . . . . . . . 12 ⊢ ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))) ∈ V
74	71, 72, 15, 73	ovmpo 7549	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))) = ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))))
76		fvco3 6960	. . . . . . . . . . . . . 14 ⊢ ((◡𝐺:𝑍⟶ω ∧ 𝑘 ∈ 𝑍) → ((𝑓 ∘ ◡𝐺)‘𝑘) = (𝑓‘(◡𝐺‘𝑘)))
77	23, 76	mpan 690	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → ((𝑓 ∘ ◡𝐺)‘𝑘) = (𝑓‘(◡𝐺‘𝑘)))
78	77	eqcomd 2735	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (𝑓‘(◡𝐺‘𝑘)) = ((𝑓 ∘ ◡𝐺)‘𝑘))
79	57, 78	oveq12d 7405	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
80	79	adantl 481	. . . . . . . . . 10 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
81	75, 80	eqtrd 2764	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
82	67, 81	eleq12d 2822	. . . . . . . 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 239	. . . . . . 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 451	. . . . . 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 3124	. . . . 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 1131	. . . 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 3451	. . . . . 6 ⊢ 𝑓 ∈ V
88		rdgfun 8384	. . . . . . . . 9 ⊢ Fun rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀)
89		omex 9596	. . . . . . . . 9 ⊢ ω ∈ V
90		resfunexg 7189	. . . . . . . . 9 ⊢ ((Fun rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω) ∈ V)
91	88, 89, 90	mp2an 692	. . . . . . . 8 ⊢ (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω) ∈ V
92	2, 91	eqeltri 2824	. . . . . . 7 ⊢ 𝐺 ∈ V
93	92	cnvex 7901	. . . . . 6 ⊢ ◡𝐺 ∈ V
94	87, 93	coex 7906	. . . . 5 ⊢ (𝑓 ∘ ◡𝐺) ∈ V
95		feq1 6666	. . . . . 6 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔:𝑍⟶𝐴 ↔ (𝑓 ∘ ◡𝐺):𝑍⟶𝐴))
96		fveq1 6857	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔‘𝑀) = ((𝑓 ∘ ◡𝐺)‘𝑀))
97	96	eqeq1d 2731	. . . . . 6 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → ((𝑔‘𝑀) = 𝐶 ↔ ((𝑓 ∘ ◡𝐺)‘𝑀) = 𝐶))
98		fveq1 6857	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔‘(𝑘 + 1)) = ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)))
99		fveq1 6857	. . . . . . . . 9 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔‘𝑘) = ((𝑓 ∘ ◡𝐺)‘𝑘))
100	99	oveq2d 7403	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑘𝐹(𝑔‘𝑘)) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
101	98, 100	eleq12d 2822	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘))))
102	101	ralbidv 3156	. . . . . 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 1438	. . . . 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 3572	. . . 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 1373	. . 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 1930	. 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 17	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∖ cdif 3911  ∅c0 4296  𝒫 cpw 4563  {csn 4589   ↦ cmpt 5188   × cxp 5636  ^◡ccnv 5637   ↾ cres 5640   ∘ ccom 5642  suc csuc 6334  Fun wfun 6505  ⟶wf 6507  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  ωcom 7842  reccrdg 8377  1c1 11069   + caddc 11071  ℤcz 12529  ℤ_≥cuz 12793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-dc 10399  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794

This theorem is referenced by:  axdc4uz  13949