Theorem axdc2lem 9716

Description: Lemma for axdc2 9717. We construct a relation 𝑅 based on 𝐹 such that 𝑥𝑅𝑦 iff 𝑦 ∈ (𝐹‘𝑥), and show that the "function" described by ax-dc 9714 can be restricted so that it is a real function (since the stated properties only show that it is the superset of a function). (Contributed by Mario Carneiro, 25-Jan-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
axdc2lem.1	⊢ 𝐴 ∈ V
axdc2lem.2	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
axdc2lem.3	⊢ 𝐺 = (𝑥 ∈ ω ↦ (ℎ‘𝑥))

Assertion

Ref	Expression
axdc2lem	⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))

Distinct variable groups:   𝐴,𝑔,ℎ   𝑥,𝐴,𝑦,ℎ   𝑔,𝐹,ℎ   𝑥,𝐹,𝑦   𝑔,𝐺,𝑘   𝑥,𝐺,𝑦,𝑘   𝑅,ℎ,𝑘,𝑥

Allowed substitution hints:   𝐴(𝑘)   𝑅(𝑦,𝑔)   𝐹(𝑘)   𝐺(ℎ)

Proof of Theorem axdc2lem

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffvelrn 6714	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
2		eldifsni 4629	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ (𝒫 𝐴 ∖ {∅}) → (𝐹‘𝑥) ≠ ∅)
3		n0 4230	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
4	2, 3	sylib 219	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
5	1, 4	syl 17	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
6	5	ralrimiva 3149	. . . . . . 7 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
7		rabid2 3340	. . . . . . 7 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
8	6, 7	sylibr 235	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → 𝐴 = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)})
9		axdc2lem.2	. . . . . . . 8 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
10	9	dmeqi 5659	. . . . . . 7 ⊢ dom 𝑅 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
11		19.42v 1931	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)))
12	11	abbii 2861	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
13		dmopab 5670	. . . . . . . 8 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
14		df-rab 3114	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
15	12, 13, 14	3eqtr4i 2829	. . . . . . 7 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)}
16	10, 15	eqtri 2819	. . . . . 6 ⊢ dom 𝑅 = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)}
17	8, 16	syl6reqr 2850	. . . . 5 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → dom 𝑅 = 𝐴)
18	17	neeq1d 3043	. . . 4 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → (dom 𝑅 ≠ ∅ ↔ 𝐴 ≠ ∅))
19	18	biimparc 480	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → dom 𝑅 ≠ ∅)
20		eldifi 4024	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ (𝒫 𝐴 ∖ {∅}) → (𝐹‘𝑥) ∈ 𝒫 𝐴)
21		elelpwi 4466	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝒫 𝐴) → 𝑦 ∈ 𝐴)
22	21	expcom 414	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ 𝒫 𝐴 → (𝑦 ∈ (𝐹‘𝑥) → 𝑦 ∈ 𝐴))
23	1, 20, 22	3syl 18	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝐹‘𝑥) → 𝑦 ∈ 𝐴))
24	23	expimpd 454	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → 𝑦 ∈ 𝐴))
25	24	exlimdv 1911	. . . . . . 7 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → 𝑦 ∈ 𝐴))
26	25	alrimiv 1905	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → 𝑦 ∈ 𝐴))
27	9	rneqi 5689	. . . . . . . . 9 ⊢ ran 𝑅 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
28		rnopab 5708	. . . . . . . . 9 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
29	27, 28	eqtri 2819	. . . . . . . 8 ⊢ ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
30	29	sseq1i 3916	. . . . . . 7 ⊢ (ran 𝑅 ⊆ 𝐴 ↔ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ⊆ 𝐴)
31		abss 3961	. . . . . . 7 ⊢ ({𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ⊆ 𝐴 ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → 𝑦 ∈ 𝐴))
32	30, 31	bitri 276	. . . . . 6 ⊢ (ran 𝑅 ⊆ 𝐴 ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → 𝑦 ∈ 𝐴))
33	26, 32	sylibr 235	. . . . 5 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ran 𝑅 ⊆ 𝐴)
34	33, 17	sseqtr4d 3929	. . . 4 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ran 𝑅 ⊆ dom 𝑅)
35	34	adantl 482	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ran 𝑅 ⊆ dom 𝑅)
36		fvrn0 6566	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) ∈ (ran 𝐹 ∪ {∅})
37		elssuni 4774	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑥) ⊆ ∪ (ran 𝐹 ∪ {∅}))
38	36, 37	ax-mp 5	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ⊆ ∪ (ran 𝐹 ∪ {∅})
39	38	sseli 3885	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐹‘𝑥) → 𝑦 ∈ ∪ (ran 𝐹 ∪ {∅}))
40	39	anim2i 616	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∪ (ran 𝐹 ∪ {∅})))
41	40	ssopab2i 5325	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∪ (ran 𝐹 ∪ {∅}))}
42		df-xp 5449	. . . . . 6 ⊢ (𝐴 × ∪ (ran 𝐹 ∪ {∅})) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∪ (ran 𝐹 ∪ {∅}))}
43	41, 9, 42	3sstr4i 3931	. . . . 5 ⊢ 𝑅 ⊆ (𝐴 × ∪ (ran 𝐹 ∪ {∅}))
44		axdc2lem.1	. . . . . 6 ⊢ 𝐴 ∈ V
45		frn 6388	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ran 𝐹 ⊆ (𝒫 𝐴 ∖ {∅}))
46	45	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ran 𝐹 ⊆ (𝒫 𝐴 ∖ {∅}))
47	44	pwex 5172	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 ∈ V
48	47	difexi 5123	. . . . . . . . . 10 ⊢ (𝒫 𝐴 ∖ {∅}) ∈ V
49	48	ssex 5116	. . . . . . . . 9 ⊢ (ran 𝐹 ⊆ (𝒫 𝐴 ∖ {∅}) → ran 𝐹 ∈ V)
50	46, 49	syl 17	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ran 𝐹 ∈ V)
51		p0ex 5175	. . . . . . . 8 ⊢ {∅} ∈ V
52		unexg 7329	. . . . . . . 8 ⊢ ((ran 𝐹 ∈ V ∧ {∅} ∈ V) → (ran 𝐹 ∪ {∅}) ∈ V)
53	50, 51, 52	sylancl 586	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → (ran 𝐹 ∪ {∅}) ∈ V)
54		uniexg 7325	. . . . . . 7 ⊢ ((ran 𝐹 ∪ {∅}) ∈ V → ∪ (ran 𝐹 ∪ {∅}) ∈ V)
55	53, 54	syl 17	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∪ (ran 𝐹 ∪ {∅}) ∈ V)
56		xpexg 7330	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∪ (ran 𝐹 ∪ {∅}) ∈ V) → (𝐴 × ∪ (ran 𝐹 ∪ {∅})) ∈ V)
57	44, 55, 56	sylancr 587	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → (𝐴 × ∪ (ran 𝐹 ∪ {∅})) ∈ V)
58		ssexg 5118	. . . . 5 ⊢ ((𝑅 ⊆ (𝐴 × ∪ (ran 𝐹 ∪ {∅})) ∧ (𝐴 × ∪ (ran 𝐹 ∪ {∅})) ∈ V) → 𝑅 ∈ V)
59	43, 57, 58	sylancr 587	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → 𝑅 ∈ V)
60		n0 4230	. . . . . . . . 9 ⊢ (dom 𝑟 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom 𝑟)
61		vex 3440	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
62	61	eldm 5655	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝑟 ↔ ∃𝑦 𝑥𝑟𝑦)
63	62	exbii 1829	. . . . . . . . 9 ⊢ (∃𝑥 𝑥 ∈ dom 𝑟 ↔ ∃𝑥∃𝑦 𝑥𝑟𝑦)
64	60, 63	bitr2i 277	. . . . . . . 8 ⊢ (∃𝑥∃𝑦 𝑥𝑟𝑦 ↔ dom 𝑟 ≠ ∅)
65		dmeq 5658	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
66	65	neeq1d 3043	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (dom 𝑟 ≠ ∅ ↔ dom 𝑅 ≠ ∅))
67	64, 66	syl5bb 284	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑥∃𝑦 𝑥𝑟𝑦 ↔ dom 𝑅 ≠ ∅))
68		rneq 5688	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
69	68, 65	sseq12d 3921	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (ran 𝑟 ⊆ dom 𝑟 ↔ ran 𝑅 ⊆ dom 𝑅))
70	67, 69	anbi12d 630	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((∃𝑥∃𝑦 𝑥𝑟𝑦 ∧ ran 𝑟 ⊆ dom 𝑟) ↔ (dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅)))
71		breq 4964	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((ℎ‘𝑘)𝑟(ℎ‘suc 𝑘) ↔ (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
72	71	ralbidv 3164	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑘 ∈ ω (ℎ‘𝑘)𝑟(ℎ‘suc 𝑘) ↔ ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
73	72	exbidv 1899	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑟(ℎ‘suc 𝑘) ↔ ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
74	70, 73	imbi12d 346	. . . . 5 ⊢ (𝑟 = 𝑅 → (((∃𝑥∃𝑦 𝑥𝑟𝑦 ∧ ran 𝑟 ⊆ dom 𝑟) → ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑟(ℎ‘suc 𝑘)) ↔ ((dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅) → ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘))))
75		ax-dc 9714	. . . . 5 ⊢ ((∃𝑥∃𝑦 𝑥𝑟𝑦 ∧ ran 𝑟 ⊆ dom 𝑟) → ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑟(ℎ‘suc 𝑘))
76	74, 75	vtoclg 3510	. . . 4 ⊢ (𝑅 ∈ V → ((dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅) → ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
77	59, 76	syl 17	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ((dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅) → ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
78	19, 35, 77	mp2and 695	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘))
79		simpr 485	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}))
80		fveq2 6538	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑥 → (ℎ‘𝑘) = (ℎ‘𝑥))
81		suceq 6131	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑥 → suc 𝑘 = suc 𝑥)
82	81	fveq2d 6542	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑥 → (ℎ‘suc 𝑘) = (ℎ‘suc 𝑥))
83	80, 82	breq12d 4975	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑥 → ((ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) ↔ (ℎ‘𝑥)𝑅(ℎ‘suc 𝑥)))
84	83	rspccv 3556	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → (𝑥 ∈ ω → (ℎ‘𝑥)𝑅(ℎ‘suc 𝑥)))
85		fvex 6551	. . . . . . . . . . . . . 14 ⊢ (ℎ‘𝑥) ∈ V
86		fvex 6551	. . . . . . . . . . . . . 14 ⊢ (ℎ‘suc 𝑥) ∈ V
87	85, 86	breldm 5663	. . . . . . . . . . . . 13 ⊢ ((ℎ‘𝑥)𝑅(ℎ‘suc 𝑥) → (ℎ‘𝑥) ∈ dom 𝑅)
88	84, 87	syl6 35	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → (𝑥 ∈ ω → (ℎ‘𝑥) ∈ dom 𝑅))
89	88	imp 407	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) ∧ 𝑥 ∈ ω) → (ℎ‘𝑥) ∈ dom 𝑅)
90	89	adantll 710	. . . . . . . . . 10 ⊢ (((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)) ∧ 𝑥 ∈ ω) → (ℎ‘𝑥) ∈ dom 𝑅)
91		eleq2 2871	. . . . . . . . . . 11 ⊢ (dom 𝑅 = 𝐴 → ((ℎ‘𝑥) ∈ dom 𝑅 ↔ (ℎ‘𝑥) ∈ 𝐴))
92	91	ad2antrr 722	. . . . . . . . . 10 ⊢ (((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)) ∧ 𝑥 ∈ ω) → ((ℎ‘𝑥) ∈ dom 𝑅 ↔ (ℎ‘𝑥) ∈ 𝐴))
93	90, 92	mpbid 233	. . . . . . . . 9 ⊢ (((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)) ∧ 𝑥 ∈ ω) → (ℎ‘𝑥) ∈ 𝐴)
94		axdc2lem.3	. . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ ω ↦ (ℎ‘𝑥))
95	93, 94	fmptd 6741	. . . . . . . 8 ⊢ ((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)) → 𝐺:ω⟶𝐴)
96	95	ex 413	. . . . . . 7 ⊢ (dom 𝑅 = 𝐴 → (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → 𝐺:ω⟶𝐴))
97	17, 96	syl 17	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → 𝐺:ω⟶𝐴))
98	97	impcom 408	. . . . 5 ⊢ ((∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → 𝐺:ω⟶𝐴)
99		fveq2 6538	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (ℎ‘𝑥) = (ℎ‘𝑘))
100		fvex 6551	. . . . . . . . . 10 ⊢ (ℎ‘𝑘) ∈ V
101	99, 94, 100	fvmpt 6635	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → (𝐺‘𝑘) = (ℎ‘𝑘))
102		peano2 7458	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
103		fvex 6551	. . . . . . . . . 10 ⊢ (ℎ‘suc 𝑘) ∈ V
104		fveq2 6538	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑘 → (ℎ‘𝑥) = (ℎ‘suc 𝑘))
105	104, 94	fvmptg 6633	. . . . . . . . . 10 ⊢ ((suc 𝑘 ∈ ω ∧ (ℎ‘suc 𝑘) ∈ V) → (𝐺‘suc 𝑘) = (ℎ‘suc 𝑘))
106	102, 103, 105	sylancl 586	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → (𝐺‘suc 𝑘) = (ℎ‘suc 𝑘))
107	101, 106	breq12d 4975	. . . . . . . 8 ⊢ (𝑘 ∈ ω → ((𝐺‘𝑘)𝑅(𝐺‘suc 𝑘) ↔ (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
108		fvex 6551	. . . . . . . . . 10 ⊢ (𝐺‘𝑘) ∈ V
109		fvex 6551	. . . . . . . . . 10 ⊢ (𝐺‘suc 𝑘) ∈ V
110		eleq1 2870	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥 ∈ 𝐴 ↔ (𝐺‘𝑘) ∈ 𝐴))
111		fveq2 6538	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑘) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑘)))
112	111	eleq2d 2868	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑦 ∈ (𝐹‘𝑥) ↔ 𝑦 ∈ (𝐹‘(𝐺‘𝑘))))
113	110, 112	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑘) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ ((𝐺‘𝑘) ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘(𝐺‘𝑘)))))
114		eleq1 2870	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘suc 𝑘) → (𝑦 ∈ (𝐹‘(𝐺‘𝑘)) ↔ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))))
115	114	anbi2d 628	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘suc 𝑘) → (((𝐺‘𝑘) ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘(𝐺‘𝑘))) ↔ ((𝐺‘𝑘) ∈ 𝐴 ∧ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘)))))
116	108, 109, 113, 115, 9	brab 5320	. . . . . . . . 9 ⊢ ((𝐺‘𝑘)𝑅(𝐺‘suc 𝑘) ↔ ((𝐺‘𝑘) ∈ 𝐴 ∧ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))))
117	116	simprbi 497	. . . . . . . 8 ⊢ ((𝐺‘𝑘)𝑅(𝐺‘suc 𝑘) → (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘)))
118	107, 117	syl6bir 255	. . . . . . 7 ⊢ (𝑘 ∈ ω → ((ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))))
119	118	ralimia 3125	. . . . . 6 ⊢ (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘)))
120	119	adantr 481	. . . . 5 ⊢ ((∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘)))
121		fvrn0 6566	. . . . . . . . . 10 ⊢ (ℎ‘𝑥) ∈ (ran ℎ ∪ {∅})
122	121	rgenw 3117	. . . . . . . . 9 ⊢ ∀𝑥 ∈ ω (ℎ‘𝑥) ∈ (ran ℎ ∪ {∅})
123		eqid 2795	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↦ (ℎ‘𝑥)) = (𝑥 ∈ ω ↦ (ℎ‘𝑥))
124	123	fmpt 6737	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ω (ℎ‘𝑥) ∈ (ran ℎ ∪ {∅}) ↔ (𝑥 ∈ ω ↦ (ℎ‘𝑥)):ω⟶(ran ℎ ∪ {∅}))
125	122, 124	mpbi 231	. . . . . . . 8 ⊢ (𝑥 ∈ ω ↦ (ℎ‘𝑥)):ω⟶(ran ℎ ∪ {∅})
126		dcomex 9715	. . . . . . . 8 ⊢ ω ∈ V
127		vex 3440	. . . . . . . . . 10 ⊢ ℎ ∈ V
128	127	rnex 7473	. . . . . . . . 9 ⊢ ran ℎ ∈ V
129	128, 51	unex 7326	. . . . . . . 8 ⊢ (ran ℎ ∪ {∅}) ∈ V
130		fex2 7494	. . . . . . . 8 ⊢ (((𝑥 ∈ ω ↦ (ℎ‘𝑥)):ω⟶(ran ℎ ∪ {∅}) ∧ ω ∈ V ∧ (ran ℎ ∪ {∅}) ∈ V) → (𝑥 ∈ ω ↦ (ℎ‘𝑥)) ∈ V)
131	125, 126, 129, 130	mp3an 1453	. . . . . . 7 ⊢ (𝑥 ∈ ω ↦ (ℎ‘𝑥)) ∈ V
132	94, 131	eqeltri 2879	. . . . . 6 ⊢ 𝐺 ∈ V
133		feq1 6363	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔:ω⟶𝐴 ↔ 𝐺:ω⟶𝐴))
134		fveq1 6537	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑔‘suc 𝑘) = (𝐺‘suc 𝑘))
135		fveq1 6537	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑘) = (𝐺‘𝑘))
136	135	fveq2d 6542	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐹‘(𝑔‘𝑘)) = (𝐹‘(𝐺‘𝑘)))
137	134, 136	eleq12d 2877	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘)) ↔ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))))
138	137	ralbidv 3164	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘)) ↔ ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))))
139	133, 138	anbi12d 630	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))) ↔ (𝐺:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘)))))
140	132, 139	spcev 3549	. . . . 5 ⊢ ((𝐺:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))
141	98, 120, 140	syl2anc 584	. . . 4 ⊢ ((∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))
142	141	ex 413	. . 3 ⊢ (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘)))))
143	142	exlimiv 1908	. 2 ⊢ (∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘)))))
144	78, 79, 143	sylc 65	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520   = wceq 1522  ∃wex 1761   ∈ wcel 2081  {cab 2775   ≠ wne 2984  ∀wral 3105  {crab 3109  Vcvv 3437   ∖ cdif 3856   ∪ cun 3857   ⊆ wss 3859  ∅c0 4211  𝒫 cpw 4453  {csn 4472  ∪ cuni 4745   class class class wbr 4962  {copab 5024   ↦ cmpt 5041   × cxp 5441  dom cdm 5443  ran crn 5444  suc csuc 6068  ⟶wf 6221  ‘cfv 6225  ωcom 7436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-dc 9714

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-om 7437  df-1o 7953

This theorem is referenced by:  axdc2  9717