Theorem ptclsg 22223

Description: The closure of a box in the product topology is the box formed from the closures of the factors. The proof uses the axiom of choice; the last hypothesis is the choice assumption. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypotheses

Ref	Expression
ptcls.2	⊢ 𝐽 = (∏t‘(𝑘 ∈ 𝐴 ↦ 𝑅))
ptcls.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ptcls.j	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑅 ∈ (TopOn‘𝑋))
ptcls.c	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ 𝑋)
ptclsg.1	⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ AC 𝐴)

Assertion

Ref	Expression
ptclsg	⊢ (𝜑 → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) = X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆))

Distinct variable groups:   𝜑,𝑘   𝐴,𝑘

Allowed substitution hints:   𝑅(𝑘)   𝑆(𝑘)   𝐽(𝑘)   𝑉(𝑘)   𝑋(𝑘)

Proof of Theorem ptclsg

Dummy variables 𝑓 𝑔 𝑢 𝑥 𝑦 𝑧 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptcls.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		ptcls.j	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑅 ∈ (TopOn‘𝑋))
3		topontop 21521	. . . . . 6 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑅 ∈ Top)
5		ptcls.c	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ 𝑋)
6		toponuni 21522	. . . . . . . 8 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
7	2, 6	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 = ∪ 𝑅)
8	5, 7	sseqtrd 4007	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ ∪ 𝑅)
9		eqid 2821	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
10	9	clscld 21655	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
11	4, 8, 10	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
12	1, 4, 11	ptcldmpt 22222	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝑅))))
13		ptcls.2	. . . . 5 ⊢ 𝐽 = (∏t‘(𝑘 ∈ 𝐴 ↦ 𝑅))
14	13	fveq2i 6673	. . . 4 ⊢ (Clsd‘𝐽) = (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝑅)))
15	12, 14	eleqtrrdi 2924	. . 3 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽))
16	9	sscls 21664	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
17	4, 8, 16	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
18	17	ralrimiva 3182	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
19		ss2ixp 8474	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆) → X𝑘 ∈ 𝐴 𝑆 ⊆ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆))
20	18, 19	syl 17	. . 3 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑆 ⊆ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆))
21		eqid 2821	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
22	21	clsss2 21680	. . 3 ⊢ ((X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽) ∧ X𝑘 ∈ 𝐴 𝑆 ⊆ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) ⊆ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆))
23	15, 20, 22	syl2anc 586	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) ⊆ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆))
24		vex 3497	. . . . . 6 ⊢ 𝑢 ∈ V
25		eqeq1 2825	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ 𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
26	25	anbi2d 630	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))))
27	26	exbidv 1922	. . . . . 6 ⊢ (𝑥 = 𝑢 → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))))
28	24, 27	elab 3667	. . . . 5 ⊢ (𝑢 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
29		nffvmpt1 6681	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)
30	29	nfel2 2996	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)
31		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(𝑔‘𝑘) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘)
32		fveq2 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑘 → (𝑔‘𝑦) = (𝑔‘𝑘))
33		fveq2 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑘 → ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) = ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘))
34	32, 33	eleq12d 2907	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → ((𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ↔ (𝑔‘𝑘) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘)))
35	30, 31, 34	cbvralw 3441	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ↔ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘))
36		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
37		eqid 2821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑅) = (𝑘 ∈ 𝐴 ↦ 𝑅)
38	37	fvmpt2 6779	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝑅 ∈ (TopOn‘𝑋)) → ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘) = 𝑅)
39	36, 2, 38	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘) = 𝑅)
40	39	eleq2d 2898	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑔‘𝑘) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘) ↔ (𝑔‘𝑘) ∈ 𝑅))
41	40	ralbidva 3196	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅))
42	35, 41	syl5bb 285	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ↔ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅))
43	42	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅)))
44	43	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → ((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅)))
45	44	biimpa 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦))) → (𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅))
46		ptclsg.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ AC 𝐴)
47	46	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ AC 𝐴)
48		simpll 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → 𝜑)
49		vex 3497	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑓 ∈ V
50	49	elixp 8468	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆)))
51	50	simprbi 499	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆))
52	51	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆))
53	9	clsndisj 21683	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ∧ (𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆)) ∧ ((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘))) → ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)
54	53	ex 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ∧ (𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆)) → (((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)) → ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅))
55	54	3expia 1117	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅) → ((𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)) → ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)))
56	4, 8, 55	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)) → ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)))
57	56	ralimdva 3177	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆) → ∀𝑘 ∈ 𝐴 (((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)) → ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)))
58	48, 52, 57	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → ∀𝑘 ∈ 𝐴 (((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)) → ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅))
59		simprlr 778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅)
60		simprr 771	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
61	32	cbvixpv 8479	. . . . . . . . . . . . . . . . . . 19 ⊢ X𝑦 ∈ 𝐴 (𝑔‘𝑦) = X𝑘 ∈ 𝐴 (𝑔‘𝑘)
62	60, 61	eleqtrdi 2923	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → 𝑓 ∈ X𝑘 ∈ 𝐴 (𝑔‘𝑘))
63	49	elixp 8468	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 (𝑔‘𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (𝑔‘𝑘)))
64	63	simprbi 499	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 (𝑔‘𝑘) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (𝑔‘𝑘))
65	62, 64	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (𝑔‘𝑘))
66		r19.26 3170	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)) ↔ (∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (𝑔‘𝑘)))
67	59, 65, 66	sylanbrc 585	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → ∀𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)))
68		ralim 3162	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝐴 (((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)) → ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅) → (∀𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)) → ∀𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅))
69	58, 67, 68	sylc 65	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → ∀𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)
70		rabn0 4339	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆)} ≠ ∅ ↔ ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆))
71		dfin5 3944	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝑘 ∈ 𝐴 𝑆 ∩ ((𝑔‘𝑘) ∩ 𝑆)) = {𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆)}
72		inss2 4206	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔‘𝑘) ∩ 𝑆) ⊆ 𝑆
73		ssiun2 4971	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝐴 → 𝑆 ⊆ ∪ 𝑘 ∈ 𝐴 𝑆)
74	72, 73	sstrid 3978	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝐴 → ((𝑔‘𝑘) ∩ 𝑆) ⊆ ∪ 𝑘 ∈ 𝐴 𝑆)
75		sseqin2 4192	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔‘𝑘) ∩ 𝑆) ⊆ ∪ 𝑘 ∈ 𝐴 𝑆 ↔ (∪ 𝑘 ∈ 𝐴 𝑆 ∩ ((𝑔‘𝑘) ∩ 𝑆)) = ((𝑔‘𝑘) ∩ 𝑆))
76	74, 75	sylib 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝐴 → (∪ 𝑘 ∈ 𝐴 𝑆 ∩ ((𝑔‘𝑘) ∩ 𝑆)) = ((𝑔‘𝑘) ∩ 𝑆))
77	71, 76	syl5eqr 2870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐴 → {𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆)} = ((𝑔‘𝑘) ∩ 𝑆))
78	77	neeq1d 3075	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝐴 → ({𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆)} ≠ ∅ ↔ ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅))
79	70, 78	syl5bbr 287	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 → (∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅))
80	79	ralbiia 3164	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ 𝐴 ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ ∀𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)
81	69, 80	sylibr 236	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → ∀𝑘 ∈ 𝐴 ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆))
82		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆)
83		nfiu1 4953	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∪ 𝑘 ∈ 𝐴 𝑆
84		nfcv 2977	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑔‘𝑦)
85		nfcsb1v 3907	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝑆
86	84, 85	nfin 4193	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)
87	86	nfel2 2996	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)
88	83, 87	nfrex 3309	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)
89		fveq2 6670	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → (𝑔‘𝑘) = (𝑔‘𝑦))
90		csbeq1a 3897	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → 𝑆 = ⦋𝑦 / 𝑘⦌𝑆)
91	89, 90	ineq12d 4190	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → ((𝑔‘𝑘) ∩ 𝑆) = ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆))
92	91	eleq2d 2898	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → (𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ 𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)))
93	92	rexbidv 3297	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑦 → (∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)))
94	82, 88, 93	cbvralw 3441	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ 𝐴 ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆))
95	81, 94	sylib 220	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆))
96		eleq1 2900	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (ℎ‘𝑦) → (𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆) ↔ (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)))
97	96	acni3 9473	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑘 ∈ 𝐴 𝑆 ∈ AC 𝐴 ∧ ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)) → ∃ℎ(ℎ:𝐴⟶∪ 𝑘 ∈ 𝐴 𝑆 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)))
98	47, 95, 97	syl2anc 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → ∃ℎ(ℎ:𝐴⟶∪ 𝑘 ∈ 𝐴 𝑆 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)))
99		ffn 6514	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐴⟶∪ 𝑘 ∈ 𝐴 𝑆 → ℎ Fn 𝐴)
100		nfv 1915	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(ℎ‘𝑘) ∈ ((𝑔‘𝑘) ∩ 𝑆)
101	86	nfel2 2996	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)
102		fveq2 6670	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → (ℎ‘𝑘) = (ℎ‘𝑦))
103	102, 91	eleq12d 2907	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → ((ℎ‘𝑘) ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)))
104	100, 101, 103	cbvralw 3441	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ 𝐴 (ℎ‘𝑘) ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆))
105		ne0i 4300	. . . . . . . . . . . . . . . 16 ⊢ (ℎ ∈ X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) → X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)
106		vex 3497	. . . . . . . . . . . . . . . . 17 ⊢ ℎ ∈ V
107	106	elixp 8468	. . . . . . . . . . . . . . . 16 ⊢ (ℎ ∈ X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) ↔ (ℎ Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (ℎ‘𝑘) ∈ ((𝑔‘𝑘) ∩ 𝑆)))
108		ixpin 8487	. . . . . . . . . . . . . . . . . 18 ⊢ X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) = (X𝑘 ∈ 𝐴 (𝑔‘𝑘) ∩ X𝑘 ∈ 𝐴 𝑆)
109	61	ineq1i 4185	. . . . . . . . . . . . . . . . . 18 ⊢ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) = (X𝑘 ∈ 𝐴 (𝑔‘𝑘) ∩ X𝑘 ∈ 𝐴 𝑆)
110	108, 109	eqtr4i 2847	. . . . . . . . . . . . . . . . 17 ⊢ X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) = (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆)
111	110	neeq1i 3080	. . . . . . . . . . . . . . . 16 ⊢ (X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅ ↔ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)
112	105, 107, 111	3imtr3i 293	. . . . . . . . . . . . . . 15 ⊢ ((ℎ Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (ℎ‘𝑘) ∈ ((𝑔‘𝑘) ∩ 𝑆)) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)
113	104, 112	sylan2br 596	. . . . . . . . . . . . . 14 ⊢ ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)
114	99, 113	sylan 582	. . . . . . . . . . . . 13 ⊢ ((ℎ:𝐴⟶∪ 𝑘 ∈ 𝐴 𝑆 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)
115	114	exlimiv 1931	. . . . . . . . . . . 12 ⊢ (∃ℎ(ℎ:𝐴⟶∪ 𝑘 ∈ 𝐴 𝑆 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)
116	98, 115	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)
117	116	expr 459	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅)) → (𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅))
118	45, 117	syldan 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦))) → (𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅))
119	118	3adantr3 1167	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦))) → (𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅))
120		eleq2 2901	. . . . . . . . 9 ⊢ (𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑓 ∈ 𝑢 ↔ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
121		ineq1 4181	. . . . . . . . . 10 ⊢ (𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) = (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆))
122	121	neeq1d 3075	. . . . . . . . 9 ⊢ (𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ((𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅ ↔ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅))
123	120, 122	imbi12d 347	. . . . . . . 8 ⊢ (𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ((𝑓 ∈ 𝑢 → (𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅) ↔ (𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)))
124	119, 123	syl5ibrcom 249	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦))) → (𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑓 ∈ 𝑢 → (𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)))
125	124	expimpd 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → (𝑓 ∈ 𝑢 → (𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)))
126	125	exlimdv 1934	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → (𝑓 ∈ 𝑢 → (𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)))
127	28, 126	syl5bi 244	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → (𝑢 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} → (𝑓 ∈ 𝑢 → (𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)))
128	127	ralrimiv 3181	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → ∀𝑢 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} (𝑓 ∈ 𝑢 → (𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅))
129	4	fmpttd 6879	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑅):𝐴⟶Top)
130	129	ffnd 6515	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑅) Fn 𝐴)
131		eqid 2821	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}
132	131	ptval 22178	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑘 ∈ 𝐴 ↦ 𝑅) Fn 𝐴) → (∏t‘(𝑘 ∈ 𝐴 ↦ 𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
133	1, 130, 132	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (∏t‘(𝑘 ∈ 𝐴 ↦ 𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
134	13, 133	syl5eq 2868	. . . . 5 ⊢ (𝜑 → 𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
135	134	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → 𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
136	2	ralrimiva 3182	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑅 ∈ (TopOn‘𝑋))
137	13	pttopon 22204	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑘 ∈ 𝐴 𝑋))
138	1, 136, 137	syl2anc 586	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘X𝑘 ∈ 𝐴 𝑋))
139		toponuni 21522	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘X𝑘 ∈ 𝐴 𝑋) → X𝑘 ∈ 𝐴 𝑋 = ∪ 𝐽)
140	138, 139	syl 17	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑋 = ∪ 𝐽)
141	140	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘 ∈ 𝐴 𝑋 = ∪ 𝐽)
142	131	ptbas 22187	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑘 ∈ 𝐴 ↦ 𝑅):𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ∈ TopBases)
143	1, 129, 142	syl2anc 586	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ∈ TopBases)
144	143	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ∈ TopBases)
145	5	ralrimiva 3182	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑆 ⊆ 𝑋)
146		ss2ixp 8474	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 𝑆 ⊆ 𝑋 → X𝑘 ∈ 𝐴 𝑆 ⊆ X𝑘 ∈ 𝐴 𝑋)
147	145, 146	syl 17	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑆 ⊆ X𝑘 ∈ 𝐴 𝑋)
148	147	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘 ∈ 𝐴 𝑆 ⊆ X𝑘 ∈ 𝐴 𝑋)
149	9	clsss3 21667	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅) → ((cls‘𝑅)‘𝑆) ⊆ ∪ 𝑅)
150	4, 8, 149	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((cls‘𝑅)‘𝑆) ⊆ ∪ 𝑅)
151	150, 7	sseqtrrd 4008	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
152	151	ralrimiva 3182	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
153		ss2ixp 8474	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋 → X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ⊆ X𝑘 ∈ 𝐴 𝑋)
154	152, 153	syl 17	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ⊆ X𝑘 ∈ 𝐴 𝑋)
155	154	sselda 3967	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓 ∈ X𝑘 ∈ 𝐴 𝑋)
156	135, 141, 144, 148, 155	elcls3 21691	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → (𝑓 ∈ ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) ↔ ∀𝑢 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} (𝑓 ∈ 𝑢 → (𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)))
157	128, 156	mpbird 259	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓 ∈ ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆))
158	23, 157	eqelssd 3988	1 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) = X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2799   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  ⦋csb 3883   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  ∪ cuni 4838  ∪ ciun 4919   ↦ cmpt 5146   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  Xcixp 8461  Fincfn 8509  AC wacn 9367  topGenctg 16711  ∏_tcpt 16712  Topctop 21501  TopOnctopon 21518  TopBasesctb 21553  Clsdccld 21624  clsccl 21626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-ixp 8462  df-en 8510  df-fin 8513  df-fi 8875  df-acn 9371  df-topgen 16717  df-pt 16718  df-top 21502  df-topon 21519  df-bases 21554  df-cld 21627  df-ntr 21628  df-cls 21629

This theorem is referenced by:  ptcls  22224  dfac14  22226