Theorem axdc2lem 9270

Description: Lemma for axdc2 9271. We construct a relation 𝑅 based on 𝐹 such that 𝑥𝑅𝑦 iff 𝑦 ∈ (𝐹‘𝑥), and show that the "function" described by ax-dc 9268 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 6357	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
2		eldifsni 4320	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ (𝒫 𝐴 ∖ {∅}) → (𝐹‘𝑥) ≠ ∅)
3		n0 3931	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
4	2, 3	sylib 208	. . . . . . . . 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 2966	. . . . . . 7 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
7		rabid2 3118	. . . . . . 7 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ (𝐹‘𝑥))
8	6, 7	sylibr 224	. . . . . 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 5325	. . . . . . 7 ⊢ dom 𝑅 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
11		19.42v 1918	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)))
12	11	abbii 2739	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
13		dmopab 5335	. . . . . . . 8 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
14		df-rab 2921	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹‘𝑥))}
15	12, 13, 14	3eqtr4i 2654	. . . . . . 7 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)}
16	10, 15	eqtri 2644	. . . . . 6 ⊢ dom 𝑅 = {𝑥 ∈ 𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹‘𝑥)}
17	8, 16	syl6reqr 2675	. . . . 5 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → dom 𝑅 = 𝐴)
18	17	neeq1d 2853	. . . 4 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → (dom 𝑅 ≠ ∅ ↔ 𝐴 ≠ ∅))
19	18	biimparc 504	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → dom 𝑅 ≠ ∅)
20		eldifi 3732	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ (𝒫 𝐴 ∖ {∅}) → (𝐹‘𝑥) ∈ 𝒫 𝐴)
21		elelpwi 4171	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝒫 𝐴) → 𝑦 ∈ 𝐴)
22	21	expcom 451	. . . . . . . . . 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 629	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → 𝑦 ∈ 𝐴))
25	24	exlimdv 1861	. . . . . . 7 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → 𝑦 ∈ 𝐴))
26	25	alrimiv 1855	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → 𝑦 ∈ 𝐴))
27	9	rneqi 5352	. . . . . . . . 9 ⊢ ran 𝑅 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
28		rnopab 5370	. . . . . . . . 9 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
29	27, 28	eqtri 2644	. . . . . . . 8 ⊢ ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
30	29	sseq1i 3629	. . . . . . 7 ⊢ (ran 𝑅 ⊆ 𝐴 ↔ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ⊆ 𝐴)
31		abss 3671	. . . . . . 7 ⊢ ({𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ⊆ 𝐴 ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → 𝑦 ∈ 𝐴))
32	30, 31	bitri 264	. . . . . 6 ⊢ (ran 𝑅 ⊆ 𝐴 ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → 𝑦 ∈ 𝐴))
33	26, 32	sylibr 224	. . . . 5 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ran 𝑅 ⊆ 𝐴)
34	33, 17	sseqtr4d 3642	. . . 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 6216	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) ∈ (ran 𝐹 ∪ {∅})
37		elssuni 4467	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑥) ⊆ ∪ (ran 𝐹 ∪ {∅}))
38	36, 37	ax-mp 5	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ⊆ ∪ (ran 𝐹 ∪ {∅})
39	38	sseli 3599	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐹‘𝑥) → 𝑦 ∈ ∪ (ran 𝐹 ∪ {∅}))
40	39	anim2i 593	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∪ (ran 𝐹 ∪ {∅})))
41	40	ssopab2i 5003	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∪ (ran 𝐹 ∪ {∅}))}
42		df-xp 5120	. . . . . 6 ⊢ (𝐴 × ∪ (ran 𝐹 ∪ {∅})) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∪ (ran 𝐹 ∪ {∅}))}
43	41, 9, 42	3sstr4i 3644	. . . . 5 ⊢ 𝑅 ⊆ (𝐴 × ∪ (ran 𝐹 ∪ {∅}))
44		axdc2lem.1	. . . . . 6 ⊢ 𝐴 ∈ V
45		frn 6053	. . . . . . . . . 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 4848	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 ∈ V
48		difexg 4808	. . . . . . . . . . 11 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∖ {∅}) ∈ V)
49	47, 48	ax-mp 5	. . . . . . . . . 10 ⊢ (𝒫 𝐴 ∖ {∅}) ∈ V
50	49	ssex 4802	. . . . . . . . 9 ⊢ (ran 𝐹 ⊆ (𝒫 𝐴 ∖ {∅}) → ran 𝐹 ∈ V)
51	46, 50	syl 17	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ran 𝐹 ∈ V)
52		p0ex 4853	. . . . . . . 8 ⊢ {∅} ∈ V
53		unexg 6959	. . . . . . . 8 ⊢ ((ran 𝐹 ∈ V ∧ {∅} ∈ V) → (ran 𝐹 ∪ {∅}) ∈ V)
54	51, 52, 53	sylancl 694	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → (ran 𝐹 ∪ {∅}) ∈ V)
55		uniexg 6955	. . . . . . 7 ⊢ ((ran 𝐹 ∪ {∅}) ∈ V → ∪ (ran 𝐹 ∪ {∅}) ∈ V)
56	54, 55	syl 17	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∪ (ran 𝐹 ∪ {∅}) ∈ V)
57		xpexg 6960	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∪ (ran 𝐹 ∪ {∅}) ∈ V) → (𝐴 × ∪ (ran 𝐹 ∪ {∅})) ∈ V)
58	44, 56, 57	sylancr 695	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → (𝐴 × ∪ (ran 𝐹 ∪ {∅})) ∈ V)
59		ssexg 4804	. . . . 5 ⊢ ((𝑅 ⊆ (𝐴 × ∪ (ran 𝐹 ∪ {∅})) ∧ (𝐴 × ∪ (ran 𝐹 ∪ {∅})) ∈ V) → 𝑅 ∈ V)
60	43, 58, 59	sylancr 695	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → 𝑅 ∈ V)
61		n0 3931	. . . . . . . . 9 ⊢ (dom 𝑟 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom 𝑟)
62		vex 3203	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
63	62	eldm 5321	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝑟 ↔ ∃𝑦 𝑥𝑟𝑦)
64	63	exbii 1774	. . . . . . . . 9 ⊢ (∃𝑥 𝑥 ∈ dom 𝑟 ↔ ∃𝑥∃𝑦 𝑥𝑟𝑦)
65	61, 64	bitr2i 265	. . . . . . . 8 ⊢ (∃𝑥∃𝑦 𝑥𝑟𝑦 ↔ dom 𝑟 ≠ ∅)
66		dmeq 5324	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
67	66	neeq1d 2853	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (dom 𝑟 ≠ ∅ ↔ dom 𝑅 ≠ ∅))
68	65, 67	syl5bb 272	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑥∃𝑦 𝑥𝑟𝑦 ↔ dom 𝑅 ≠ ∅))
69		rneq 5351	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
70	69, 66	sseq12d 3634	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (ran 𝑟 ⊆ dom 𝑟 ↔ ran 𝑅 ⊆ dom 𝑅))
71	68, 70	anbi12d 747	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((∃𝑥∃𝑦 𝑥𝑟𝑦 ∧ ran 𝑟 ⊆ dom 𝑟) ↔ (dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅)))
72		breq 4655	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((ℎ‘𝑘)𝑟(ℎ‘suc 𝑘) ↔ (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
73	72	ralbidv 2986	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑘 ∈ ω (ℎ‘𝑘)𝑟(ℎ‘suc 𝑘) ↔ ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
74	73	exbidv 1850	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑟(ℎ‘suc 𝑘) ↔ ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
75	71, 74	imbi12d 334	. . . . 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))))
76		ax-dc 9268	. . . . 5 ⊢ ((∃𝑥∃𝑦 𝑥𝑟𝑦 ∧ ran 𝑟 ⊆ dom 𝑟) → ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑟(ℎ‘suc 𝑘))
77	75, 76	vtoclg 3266	. . . 4 ⊢ (𝑅 ∈ V → ((dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅) → ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
78	60, 77	syl 17	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ((dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅) → ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
79	19, 35, 78	mp2and 715	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃ℎ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘))
80		simpr 477	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}))
81		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑥 → (ℎ‘𝑘) = (ℎ‘𝑥))
82		suceq 5790	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑥 → suc 𝑘 = suc 𝑥)
83	82	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑥 → (ℎ‘suc 𝑘) = (ℎ‘suc 𝑥))
84	81, 83	breq12d 4666	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑥 → ((ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) ↔ (ℎ‘𝑥)𝑅(ℎ‘suc 𝑥)))
85	84	rspccv 3306	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → (𝑥 ∈ ω → (ℎ‘𝑥)𝑅(ℎ‘suc 𝑥)))
86		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (ℎ‘𝑥) ∈ V
87		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (ℎ‘suc 𝑥) ∈ V
88	86, 87	breldm 5329	. . . . . . . . . . . . 13 ⊢ ((ℎ‘𝑥)𝑅(ℎ‘suc 𝑥) → (ℎ‘𝑥) ∈ dom 𝑅)
89	85, 88	syl6 35	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → (𝑥 ∈ ω → (ℎ‘𝑥) ∈ dom 𝑅))
90	89	imp 445	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) ∧ 𝑥 ∈ ω) → (ℎ‘𝑥) ∈ dom 𝑅)
91	90	adantll 750	. . . . . . . . . 10 ⊢ (((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)) ∧ 𝑥 ∈ ω) → (ℎ‘𝑥) ∈ dom 𝑅)
92		eleq2 2690	. . . . . . . . . . 11 ⊢ (dom 𝑅 = 𝐴 → ((ℎ‘𝑥) ∈ dom 𝑅 ↔ (ℎ‘𝑥) ∈ 𝐴))
93	92	ad2antrr 762	. . . . . . . . . 10 ⊢ (((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)) ∧ 𝑥 ∈ ω) → ((ℎ‘𝑥) ∈ dom 𝑅 ↔ (ℎ‘𝑥) ∈ 𝐴))
94	91, 93	mpbid 222	. . . . . . . . 9 ⊢ (((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)) ∧ 𝑥 ∈ ω) → (ℎ‘𝑥) ∈ 𝐴)
95		axdc2lem.3	. . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ ω ↦ (ℎ‘𝑥))
96	94, 95	fmptd 6385	. . . . . . . 8 ⊢ ((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)) → 𝐺:ω⟶𝐴)
97	96	ex 450	. . . . . . 7 ⊢ (dom 𝑅 = 𝐴 → (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → 𝐺:ω⟶𝐴))
98	17, 97	syl 17	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → 𝐺:ω⟶𝐴))
99	98	impcom 446	. . . . 5 ⊢ ((∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → 𝐺:ω⟶𝐴)
100		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (ℎ‘𝑥) = (ℎ‘𝑘))
101		fvex 6201	. . . . . . . . . 10 ⊢ (ℎ‘𝑘) ∈ V
102	100, 95, 101	fvmpt 6282	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → (𝐺‘𝑘) = (ℎ‘𝑘))
103		peano2 7086	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
104		fvex 6201	. . . . . . . . . 10 ⊢ (ℎ‘suc 𝑘) ∈ V
105		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑘 → (ℎ‘𝑥) = (ℎ‘suc 𝑘))
106	105, 95	fvmptg 6280	. . . . . . . . . 10 ⊢ ((suc 𝑘 ∈ ω ∧ (ℎ‘suc 𝑘) ∈ V) → (𝐺‘suc 𝑘) = (ℎ‘suc 𝑘))
107	103, 104, 106	sylancl 694	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → (𝐺‘suc 𝑘) = (ℎ‘suc 𝑘))
108	102, 107	breq12d 4666	. . . . . . . 8 ⊢ (𝑘 ∈ ω → ((𝐺‘𝑘)𝑅(𝐺‘suc 𝑘) ↔ (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘)))
109		fvex 6201	. . . . . . . . . 10 ⊢ (𝐺‘𝑘) ∈ V
110		fvex 6201	. . . . . . . . . 10 ⊢ (𝐺‘suc 𝑘) ∈ V
111		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥 ∈ 𝐴 ↔ (𝐺‘𝑘) ∈ 𝐴))
112		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑘) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑘)))
113	112	eleq2d 2687	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑦 ∈ (𝐹‘𝑥) ↔ 𝑦 ∈ (𝐹‘(𝐺‘𝑘))))
114	111, 113	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑘) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ ((𝐺‘𝑘) ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘(𝐺‘𝑘)))))
115		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘suc 𝑘) → (𝑦 ∈ (𝐹‘(𝐺‘𝑘)) ↔ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))))
116	115	anbi2d 740	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘suc 𝑘) → (((𝐺‘𝑘) ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘(𝐺‘𝑘))) ↔ ((𝐺‘𝑘) ∈ 𝐴 ∧ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘)))))
117	109, 110, 114, 116, 9	brab 4998	. . . . . . . . 9 ⊢ ((𝐺‘𝑘)𝑅(𝐺‘suc 𝑘) ↔ ((𝐺‘𝑘) ∈ 𝐴 ∧ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))))
118	117	simprbi 480	. . . . . . . 8 ⊢ ((𝐺‘𝑘)𝑅(𝐺‘suc 𝑘) → (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘)))
119	108, 118	syl6bir 244	. . . . . . 7 ⊢ (𝑘 ∈ ω → ((ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))))
120	119	ralimia 2950	. . . . . 6 ⊢ (∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) → ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘)))
121	120	adantr 481	. . . . 5 ⊢ ((∀𝑘 ∈ ω (ℎ‘𝑘)𝑅(ℎ‘suc 𝑘) ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘)))
122		fvrn0 6216	. . . . . . . . . 10 ⊢ (ℎ‘𝑥) ∈ (ran ℎ ∪ {∅})
123	122	rgenw 2924	. . . . . . . . 9 ⊢ ∀𝑥 ∈ ω (ℎ‘𝑥) ∈ (ran ℎ ∪ {∅})
124		eqid 2622	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↦ (ℎ‘𝑥)) = (𝑥 ∈ ω ↦ (ℎ‘𝑥))
125	124	fmpt 6381	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ω (ℎ‘𝑥) ∈ (ran ℎ ∪ {∅}) ↔ (𝑥 ∈ ω ↦ (ℎ‘𝑥)):ω⟶(ran ℎ ∪ {∅}))
126	123, 125	mpbi 220	. . . . . . . 8 ⊢ (𝑥 ∈ ω ↦ (ℎ‘𝑥)):ω⟶(ran ℎ ∪ {∅})
127		dcomex 9269	. . . . . . . 8 ⊢ ω ∈ V
128		vex 3203	. . . . . . . . . 10 ⊢ ℎ ∈ V
129	128	rnex 7100	. . . . . . . . 9 ⊢ ran ℎ ∈ V
130	129, 52	unex 6956	. . . . . . . 8 ⊢ (ran ℎ ∪ {∅}) ∈ V
131		fex2 7121	. . . . . . . 8 ⊢ (((𝑥 ∈ ω ↦ (ℎ‘𝑥)):ω⟶(ran ℎ ∪ {∅}) ∧ ω ∈ V ∧ (ran ℎ ∪ {∅}) ∈ V) → (𝑥 ∈ ω ↦ (ℎ‘𝑥)) ∈ V)
132	126, 127, 130, 131	mp3an 1424	. . . . . . 7 ⊢ (𝑥 ∈ ω ↦ (ℎ‘𝑥)) ∈ V
133	95, 132	eqeltri 2697	. . . . . 6 ⊢ 𝐺 ∈ V
134		feq1 6026	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔:ω⟶𝐴 ↔ 𝐺:ω⟶𝐴))
135		fveq1 6190	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑔‘suc 𝑘) = (𝐺‘suc 𝑘))
136		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑘) = (𝐺‘𝑘))
137	136	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐹‘(𝑔‘𝑘)) = (𝐹‘(𝐺‘𝑘)))
138	135, 137	eleq12d 2695	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘)) ↔ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))))
139	138	ralbidv 2986	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘)) ↔ ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))))
140	134, 139	anbi12d 747	. . . . . 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)))))
141	133, 140	spcev 3300	. . . . 5 ⊢ ((𝐺:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺‘𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))
142	99, 121, 141	syl2anc 693	. . . 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))))
143	142	ex 450	. . 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)))))
144	143	exlimiv 1858	. 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)))))
145	79, 80, 144	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 196   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ran crn 5115  suc csuc 5725  ⟶wf 5884  ‘cfv 5888  ωcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-dc 9268

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-om 7066  df-1o 7560

This theorem is referenced by:  axdc2  9271