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Theorem ptclsg 22220

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 21518	. . . . . 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 21519	. . . . . . . 8 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
7	2, 6	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 = ∪ 𝑅)
8	5, 7	sseqtrd 3955	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ ∪ 𝑅)
9		eqid 2798	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
10	9	clscld 21652	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
11	4, 8, 10	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
12	1, 4, 11	ptcldmpt 22219	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘(∏_t‘(𝑘 ∈ 𝐴 ↦ 𝑅))))
13		ptcls.2	. . . . 5 ⊢ 𝐽 = (∏_t‘(𝑘 ∈ 𝐴 ↦ 𝑅))
14	13	fveq2i 6648	. . . 4 ⊢ (Clsd‘𝐽) = (Clsd‘(∏_t‘(𝑘 ∈ 𝐴 ↦ 𝑅)))
15	12, 14	eleqtrrdi 2901	. . 3 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽))
16	9	sscls 21661	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
17	4, 8, 16	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
18	17	ralrimiva 3149	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
19		ss2ixp 8457	. . . 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 2798	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
22	21	clsss2 21677	. . 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 587	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) ⊆ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆))
24		vex 3444	. . . . . 6 ⊢ 𝑢 ∈ V
25		eqeq1 2802	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ 𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
26	25	anbi2d 631	. . . . . . 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 3615	. . . . 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 6656	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)
30	29	nfel2 2973	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)
31		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(𝑔‘𝑘) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘)
32		fveq2 6645	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑘 → (𝑔‘𝑦) = (𝑔‘𝑘))
33		fveq2 6645	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑘 → ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) = ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘))
34	32, 33	eleq12d 2884	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → ((𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ↔ (𝑔‘𝑘) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘)))
35	30, 31, 34	cbvralw 3387	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ↔ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘))
36		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
37		eqid 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑅) = (𝑘 ∈ 𝐴 ↦ 𝑅)
38	37	fvmpt2 6756	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝑅 ∈ (TopOn‘𝑋)) → ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘) = 𝑅)
39	36, 2, 38	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘) = 𝑅)
40	39	eleq2d 2875	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑔‘𝑘) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘) ↔ (𝑔‘𝑘) ∈ 𝑅))
41	40	ralbidva 3161	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅))
42	35, 41	syl5bb 286	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦) ↔ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅))
43	42	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝑘 ∈ 𝐴 ↦ 𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅)))
44	43	adantr 484	. . . . . . . . . . 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 480	. . . . . . . . . 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 725	. . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ 𝑅) ∧ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))) → 𝜑)
49		vex 3444	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑓 ∈ V
50	49	elixp 8451	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆)))
51	50	simprbi 500	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆))
52	51	ad2antlr 726	. . . . . . . . . . . . . . . . 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 21680	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ∧ (𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆)) ∧ ((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘))) → ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)
54	53	ex 416	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ∧ (𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆)) → (((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)) → ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅))
55	54	3expia 1118	. . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑓‘𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔‘𝑘) ∈ 𝑅 ∧ (𝑓‘𝑘) ∈ (𝑔‘𝑘)) → ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)))
57	56	ralimdva 3144	. . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . 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 8462	. . . . . . . . . . . . . . . . . . 19 ⊢ X𝑦 ∈ 𝐴 (𝑔‘𝑦) = X𝑘 ∈ 𝐴 (𝑔‘𝑘)
62	60, 61	eleqtrdi 2900	. . . . . . . . . . . . . . . . . 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 8451	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 (𝑔‘𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (𝑔‘𝑘)))
64	63	simprbi 500	. . . . . . . . . . . . . . . . . 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 3137	. . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . 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 3130	. . . . . . . . . . . . . . . 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 4293	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆)} ≠ ∅ ↔ ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆))
71		dfin5 3889	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝑘 ∈ 𝐴 𝑆 ∩ ((𝑔‘𝑘) ∩ 𝑆)) = {𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆)}
72		inss2 4156	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔‘𝑘) ∩ 𝑆) ⊆ 𝑆
73		ssiun2 4934	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝐴 → 𝑆 ⊆ ∪ 𝑘 ∈ 𝐴 𝑆)
74	72, 73	sstrid 3926	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝐴 → ((𝑔‘𝑘) ∩ 𝑆) ⊆ ∪ 𝑘 ∈ 𝐴 𝑆)
75		sseqin2 4142	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔‘𝑘) ∩ 𝑆) ⊆ ∪ 𝑘 ∈ 𝐴 𝑆 ↔ (∪ 𝑘 ∈ 𝐴 𝑆 ∩ ((𝑔‘𝑘) ∩ 𝑆)) = ((𝑔‘𝑘) ∩ 𝑆))
76	74, 75	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝐴 → (∪ 𝑘 ∈ 𝐴 𝑆 ∩ ((𝑔‘𝑘) ∩ 𝑆)) = ((𝑔‘𝑘) ∩ 𝑆))
77	71, 76	syl5eqr 2847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐴 → {𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆)} = ((𝑔‘𝑘) ∩ 𝑆))
78	77	neeq1d 3046	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝐴 → ({𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆)} ≠ ∅ ↔ ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅))
79	70, 78	bitr3id 288	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 → (∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅))
80	79	ralbiia 3132	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ 𝐴 ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ ∀𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)
81	69, 80	sylibr 237	. . . . . . . . . . . . . 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 4915	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∪ 𝑘 ∈ 𝐴 𝑆
84		nfcv 2955	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑔‘𝑦)
85		nfcsb1v 3852	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝑆
86	84, 85	nfin 4143	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)
87	86	nfel2 2973	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)
88	83, 87	nfrex 3268	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)
89		fveq2 6645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → (𝑔‘𝑘) = (𝑔‘𝑦))
90		csbeq1a 3842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → 𝑆 = ⦋𝑦 / 𝑘⦌𝑆)
91	89, 90	ineq12d 4140	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → ((𝑔‘𝑘) ∩ 𝑆) = ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆))
92	91	eleq2d 2875	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → (𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ 𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)))
93	92	rexbidv 3256	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑦 → (∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ ∃𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)))
94	82, 88, 93	cbvralw 3387	. . . . . . . . . . . . . 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 221	. . . . . . . . . . . . 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 2877	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (ℎ‘𝑦) → (𝑧 ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆) ↔ (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)))
97	96	acni3 9458	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 6487	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐴⟶∪ 𝑘 ∈ 𝐴 𝑆 → ℎ Fn 𝐴)
100		nfv 1915	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(ℎ‘𝑘) ∈ ((𝑔‘𝑘) ∩ 𝑆)
101	86	nfel2 2973	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)
102		fveq2 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → (ℎ‘𝑘) = (ℎ‘𝑦))
103	102, 91	eleq12d 2884	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → ((ℎ‘𝑘) ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)))
104	100, 101, 103	cbvralw 3387	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ 𝐴 (ℎ‘𝑘) ∈ ((𝑔‘𝑘) ∩ 𝑆) ↔ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆))
105		ne0i 4250	. . . . . . . . . . . . . . . 16 ⊢ (ℎ ∈ X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) → X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅)
106		vex 3444	. . . . . . . . . . . . . . . . 17 ⊢ ℎ ∈ V
107	106	elixp 8451	. . . . . . . . . . . . . . . 16 ⊢ (ℎ ∈ X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) ↔ (ℎ Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (ℎ‘𝑘) ∈ ((𝑔‘𝑘) ∩ 𝑆)))
108		ixpin 8470	. . . . . . . . . . . . . . . . . 18 ⊢ X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) = (X𝑘 ∈ 𝐴 (𝑔‘𝑘) ∩ X𝑘 ∈ 𝐴 𝑆)
109	61	ineq1i 4135	. . . . . . . . . . . . . . . . . 18 ⊢ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) = (X𝑘 ∈ 𝐴 (𝑔‘𝑘) ∩ X𝑘 ∈ 𝐴 𝑆)
110	108, 109	eqtr4i 2824	. . . . . . . . . . . . . . . . 17 ⊢ X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) = (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆)
111	110	neeq1i 3051	. . . . . . . . . . . . . . . 16 ⊢ (X𝑘 ∈ 𝐴 ((𝑔‘𝑘) ∩ 𝑆) ≠ ∅ ↔ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)
112	105, 107, 111	3imtr3i 294	. . . . . . . . . . . . . . 15 ⊢ ((ℎ Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (ℎ‘𝑘) ∈ ((𝑔‘𝑘) ∩ 𝑆)) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)
113	104, 112	sylan2br 597	. . . . . . . . . . . . . 14 ⊢ ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ ((𝑔‘𝑦) ∩ ⦋𝑦 / 𝑘⦌𝑆)) → (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅)
114	99, 113	sylan 583	. . . . . . . . . . . . 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 460	. . . . . . . . . 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 594	. . . . . . . . 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 1168	. . . . . . . 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 2878	. . . . . . . . 9 ⊢ (𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑓 ∈ 𝑢 ↔ 𝑓 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
121		ineq1 4131	. . . . . . . . . 10 ⊢ (𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) = (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆))
122	121	neeq1d 3046	. . . . . . . . 9 ⊢ (𝑢 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ((𝑢 ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅ ↔ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∩ X𝑘 ∈ 𝐴 𝑆) ≠ ∅))
123	120, 122	imbi12d 348	. . . . . . . 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 250	. . . . . . 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 457	. . . . . 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 245	. . . 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 3148	. . 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 6856	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑅):𝐴⟶Top)
130	129	ffnd 6488	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑅) Fn 𝐴)
131		eqid 2798	. . . . . . . 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 22175	. . . . . . 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 587	. . . . . 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 2845	. . . . 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 484	. . . 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 3149	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑅 ∈ (TopOn‘𝑋))
137	13	pttopon 22201	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑘 ∈ 𝐴 𝑋))
138	1, 136, 137	syl2anc 587	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘X𝑘 ∈ 𝐴 𝑋))
139		toponuni 21519	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘X𝑘 ∈ 𝐴 𝑋) → X𝑘 ∈ 𝐴 𝑋 = ∪ 𝐽)
140	138, 139	syl 17	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑋 = ∪ 𝐽)
141	140	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘 ∈ 𝐴 𝑋 = ∪ 𝐽)
142	131	ptbas 22184	. . . . . 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 587	. . . . 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 484	. . . 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 3149	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑆 ⊆ 𝑋)
146		ss2ixp 8457	. . . . . 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘 ∈ 𝐴 𝑆 ⊆ X𝑘 ∈ 𝐴 𝑋)
149	9	clsss3 21664	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅) → ((cls‘𝑅)‘𝑆) ⊆ ∪ 𝑅)
150	4, 8, 149	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((cls‘𝑅)‘𝑆) ⊆ ∪ 𝑅)
151	150, 7	sseqtrrd 3956	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
152	151	ralrimiva 3149	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
153		ss2ixp 8457	. . . . . 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 3915	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓 ∈ X𝑘 ∈ 𝐴 𝑋)
156	135, 141, 144, 148, 155	elcls3 21688	. . 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 260	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓 ∈ ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆))
158	23, 157	eqelssd 3936	1 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) = X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 {crab 3110 ⦋csb 3828 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 ∪ ciun 4881 ↦ cmpt 5110 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 Xcixp 8444 Fincfn 8492 AC wacn 9351 topGenctg 16703 ∏_tcpt 16704 Topctop 21498 TopOnctopon 21515 TopBasesctb 21550 Clsdccld 21621 clsccl 21623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-fin 8496 df-fi 8859 df-acn 9355 df-topgen 16709 df-pt 16710 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626

This theorem is referenced by: ptcls 22221 dfac14 22223

W3C validator