Theorem axdc4uzlem 13105

Description: Lemma for axdc4uz 13106. (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 13075	. . . . . . . . . 10 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝑀)
4		axdc4uz.2	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ≥‘𝑀)
5		f1oeq3 6384	. . . . . . . . . . 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 223	. . . . . . . . 9 ⊢ 𝐺:ω–1-1-onto→𝑍
8		f1of 6393	. . . . . . . . 9 ⊢ (𝐺:ω–1-1-onto→𝑍 → 𝐺:ω⟶𝑍)
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ 𝐺:ω⟶𝑍
10	9	ffvelrni 6624	. . . . . . 7 ⊢ (𝑛 ∈ ω → (𝐺‘𝑛) ∈ 𝑍)
11		fovrn 7083	. . . . . . 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 1110	. . . . 5 ⊢ ((𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ 𝐴)) → ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
14	13	ralrimivva 3153	. . . 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	fmpt2 7519	. . . 4 ⊢ (∀𝑛 ∈ ω ∀𝑥 ∈ 𝐴 ((𝐺‘𝑛)𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}) ↔ 𝐻:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅}))
17	14, 16	sylib 210	. . 3 ⊢ (𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}) → 𝐻:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅}))
18		axdc4uz.3	. . . 4 ⊢ 𝐴 ∈ V
19	18	axdc4 9615	. . 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 586	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑓(𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))))
21		f1ocnv 6405	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→𝑍 → ◡𝐺:𝑍–1-1-onto→ω)
22		f1of 6393	. . . . . . 7 ⊢ (◡𝐺:𝑍–1-1-onto→ω → ◡𝐺:𝑍⟶ω)
23	7, 21, 22	mp2b 10	. . . . . 6 ⊢ ◡𝐺:𝑍⟶ω
24		fco 6310	. . . . . 6 ⊢ ((𝑓:ω⟶𝐴 ∧ ◡𝐺:𝑍⟶ω) → (𝑓 ∘ ◡𝐺):𝑍⟶𝐴)
25	23, 24	mpan2 681	. . . . 5 ⊢ (𝑓:ω⟶𝐴 → (𝑓 ∘ ◡𝐺):𝑍⟶𝐴)
26	25	3ad2ant1 1124	. . . 4 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → (𝑓 ∘ ◡𝐺):𝑍⟶𝐴)
27		uzid 12011	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
28	1, 27	ax-mp 5	. . . . . . . 8 ⊢ 𝑀 ∈ (ℤ≥‘𝑀)
29	28, 4	eleqtrri 2858	. . . . . . 7 ⊢ 𝑀 ∈ 𝑍
30		fvco3 6537	. . . . . . 7 ⊢ ((◡𝐺:𝑍⟶ω ∧ 𝑀 ∈ 𝑍) → ((𝑓 ∘ ◡𝐺)‘𝑀) = (𝑓‘(◡𝐺‘𝑀)))
31	23, 29, 30	mp2an 682	. . . . . 6 ⊢ ((𝑓 ∘ ◡𝐺)‘𝑀) = (𝑓‘(◡𝐺‘𝑀))
32	1, 2	om2uz0i 13069	. . . . . . . 8 ⊢ (𝐺‘∅) = 𝑀
33		peano1 7365	. . . . . . . . 9 ⊢ ∅ ∈ ω
34		f1ocnvfv 6808	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ ∅ ∈ ω) → ((𝐺‘∅) = 𝑀 → (◡𝐺‘𝑀) = ∅))
35	7, 33, 34	mp2an 682	. . . . . . . 8 ⊢ ((𝐺‘∅) = 𝑀 → (◡𝐺‘𝑀) = ∅)
36	32, 35	ax-mp 5	. . . . . . 7 ⊢ (◡𝐺‘𝑀) = ∅
37	36	fveq2i 6451	. . . . . 6 ⊢ (𝑓‘(◡𝐺‘𝑀)) = (𝑓‘∅)
38	31, 37	eqtri 2802	. . . . 5 ⊢ ((𝑓 ∘ ◡𝐺)‘𝑀) = (𝑓‘∅)
39		simp2 1128	. . . . 5 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → (𝑓‘∅) = 𝐶)
40	38, 39	syl5eq 2826	. . . 4 ⊢ ((𝑓:ω⟶𝐴 ∧ (𝑓‘∅) = 𝐶 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚))) → ((𝑓 ∘ ◡𝐺)‘𝑀) = 𝐶)
41	23	ffvelrni 6624	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 → (◡𝐺‘𝑘) ∈ ω)
42	41	adantl 475	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (◡𝐺‘𝑘) ∈ ω)
43		suceq 6043	. . . . . . . . . . . 12 ⊢ (𝑚 = (◡𝐺‘𝑘) → suc 𝑚 = suc (◡𝐺‘𝑘))
44	43	fveq2d 6452	. . . . . . . . . . 11 ⊢ (𝑚 = (◡𝐺‘𝑘) → (𝑓‘suc 𝑚) = (𝑓‘suc (◡𝐺‘𝑘)))
45		id 22	. . . . . . . . . . . 12 ⊢ (𝑚 = (◡𝐺‘𝑘) → 𝑚 = (◡𝐺‘𝑘))
46		fveq2 6448	. . . . . . . . . . . 12 ⊢ (𝑚 = (◡𝐺‘𝑘) → (𝑓‘𝑚) = (𝑓‘(◡𝐺‘𝑘)))
47	45, 46	oveq12d 6942	. . . . . . . . . . 11 ⊢ (𝑚 = (◡𝐺‘𝑘) → (𝑚𝐻(𝑓‘𝑚)) = ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))))
48	44, 47	eleq12d 2853	. . . . . . . . . 10 ⊢ (𝑚 = (◡𝐺‘𝑘) → ((𝑓‘suc 𝑚) ∈ (𝑚𝐻(𝑓‘𝑚)) ↔ (𝑓‘suc (◡𝐺‘𝑘)) ∈ ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘)))))
49	48	rspcv 3507	. . . . . . . . 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 12052	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (𝑘 + 1) ∈ 𝑍)
52		fvco3 6537	. . . . . . . . . . . 12 ⊢ ((◡𝐺:𝑍⟶ω ∧ (𝑘 + 1) ∈ 𝑍) → ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) = (𝑓‘(◡𝐺‘(𝑘 + 1))))
53	23, 51, 52	sylancr 581	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) = (𝑓‘(◡𝐺‘(𝑘 + 1))))
54	1, 2	om2uzsuci 13070	. . . . . . . . . . . . . . 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 6807	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝑘)) = 𝑘)
57	7, 56	mpan 680	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘(◡𝐺‘𝑘)) = 𝑘)
58	57	oveq1d 6939	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 → ((𝐺‘(◡𝐺‘𝑘)) + 1) = (𝑘 + 1))
59	55, 58	eqtrd 2814	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘suc (◡𝐺‘𝑘)) = (𝑘 + 1))
60		peano2 7366	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐺‘𝑘) ∈ ω → suc (◡𝐺‘𝑘) ∈ ω)
61	41, 60	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 → suc (◡𝐺‘𝑘) ∈ ω)
62		f1ocnvfv 6808	. . . . . . . . . . . . . 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 581	. . . . . . . . . . . . 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 6452	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → (𝑓‘(◡𝐺‘(𝑘 + 1))) = (𝑓‘suc (◡𝐺‘𝑘)))
66	53, 65	eqtr2d 2815	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 → (𝑓‘suc (◡𝐺‘𝑘)) = ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)))
67	66	adantl 475	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (𝑓‘suc (◡𝐺‘𝑘)) = ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)))
68		ffvelrn 6623	. . . . . . . . . . . 12 ⊢ ((𝑓:ω⟶𝐴 ∧ (◡𝐺‘𝑘) ∈ ω) → (𝑓‘(◡𝐺‘𝑘)) ∈ 𝐴)
69	41, 68	sylan2 586	. . . . . . . . . . 11 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → (𝑓‘(◡𝐺‘𝑘)) ∈ 𝐴)
70		fveq2 6448	. . . . . . . . . . . . 13 ⊢ (𝑛 = (◡𝐺‘𝑘) → (𝐺‘𝑛) = (𝐺‘(◡𝐺‘𝑘)))
71	70	oveq1d 6939	. . . . . . . . . . . 12 ⊢ (𝑛 = (◡𝐺‘𝑘) → ((𝐺‘𝑛)𝐹𝑥) = ((𝐺‘(◡𝐺‘𝑘))𝐹𝑥))
72		oveq2 6932	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘(◡𝐺‘𝑘)) → ((𝐺‘(◡𝐺‘𝑘))𝐹𝑥) = ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))))
73		ovex 6956	. . . . . . . . . . . 12 ⊢ ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))) ∈ V
74	71, 72, 15, 73	ovmpt2 7075	. . . . . . . . . . 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 579	. . . . . . . . . 10 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))) = ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))))
76		fvco3 6537	. . . . . . . . . . . . . 14 ⊢ ((◡𝐺:𝑍⟶ω ∧ 𝑘 ∈ 𝑍) → ((𝑓 ∘ ◡𝐺)‘𝑘) = (𝑓‘(◡𝐺‘𝑘)))
77	23, 76	mpan 680	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → ((𝑓 ∘ ◡𝐺)‘𝑘) = (𝑓‘(◡𝐺‘𝑘)))
78	77	eqcomd 2784	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (𝑓‘(◡𝐺‘𝑘)) = ((𝑓 ∘ ◡𝐺)‘𝑘))
79	57, 78	oveq12d 6942	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
80	79	adantl 475	. . . . . . . . . 10 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘(◡𝐺‘𝑘))𝐹(𝑓‘(◡𝐺‘𝑘))) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
81	75, 80	eqtrd 2814	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((◡𝐺‘𝑘)𝐻(𝑓‘(◡𝐺‘𝑘))) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
82	67, 81	eleq12d 2853	. . . . . . . 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 231	. . . . . . 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 445	. . . . . 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 3147	. . . . 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 1122	. . . 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 3401	. . . . . 6 ⊢ 𝑓 ∈ V
88		rdgfun 7797	. . . . . . . . 9 ⊢ Fun rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀)
89		omex 8839	. . . . . . . . 9 ⊢ ω ∈ V
90		resfunexg 6753	. . . . . . . . 9 ⊢ ((Fun rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω) ∈ V)
91	88, 89, 90	mp2an 682	. . . . . . . 8 ⊢ (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω) ∈ V
92	2, 91	eqeltri 2855	. . . . . . 7 ⊢ 𝐺 ∈ V
93	92	cnvex 7394	. . . . . 6 ⊢ ◡𝐺 ∈ V
94	87, 93	coex 7399	. . . . 5 ⊢ (𝑓 ∘ ◡𝐺) ∈ V
95		feq1 6274	. . . . . 6 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔:𝑍⟶𝐴 ↔ (𝑓 ∘ ◡𝐺):𝑍⟶𝐴))
96		fveq1 6447	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔‘𝑀) = ((𝑓 ∘ ◡𝐺)‘𝑀))
97	96	eqeq1d 2780	. . . . . 6 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → ((𝑔‘𝑀) = 𝐶 ↔ ((𝑓 ∘ ◡𝐺)‘𝑀) = 𝐶))
98		fveq1 6447	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔‘(𝑘 + 1)) = ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)))
99		fveq1 6447	. . . . . . . . 9 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑔‘𝑘) = ((𝑓 ∘ ◡𝐺)‘𝑘))
100	99	oveq2d 6940	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → (𝑘𝐹(𝑔‘𝑘)) = (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘)))
101	98, 100	eleq12d 2853	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∘ ◡𝐺) → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ ((𝑓 ∘ ◡𝐺)‘(𝑘 + 1)) ∈ (𝑘𝐹((𝑓 ∘ ◡𝐺)‘𝑘))))
102	101	ralbidv 3168	. . . . . 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 1509	. . . . 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 3502	. . . 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 1439	. . 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 1973	. 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 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601  ∃wex 1823   ∈ wcel 2107  ∀wral 3090  Vcvv 3398   ∖ cdif 3789  ∅c0 4141  𝒫 cpw 4379  {csn 4398   ↦ cmpt 4967   × cxp 5355  ^◡ccnv 5356   ↾ cres 5359   ∘ ccom 5361  suc csuc 5980  Fun wfun 6131  ⟶wf 6133  –1-1-onto→wf1o 6136  ‘cfv 6137  (class class class)co 6924   ↦ cmpt2 6926  ωcom 7345  reccrdg 7790  1c1 10275   + caddc 10277  ℤcz 11732  ℤ_≥cuz 11996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-dc 9605  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11379  df-n0 11647  df-z 11733  df-uz 11997

This theorem is referenced by:  axdc4uz  13106