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Theorem ptclsg 21698
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‘(𝑘𝐴𝑅))
ptcls.a (𝜑𝐴𝑉)
ptcls.j ((𝜑𝑘𝐴) → 𝑅 ∈ (TopOn‘𝑋))
ptcls.c ((𝜑𝑘𝐴) → 𝑆𝑋)
ptclsg.1 (𝜑 𝑘𝐴 𝑆AC 𝐴)
Assertion
Ref Expression
ptclsg (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) = X𝑘𝐴 ((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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcls.a . . . . 5 (𝜑𝐴𝑉)
2 ptcls.j . . . . . 6 ((𝜑𝑘𝐴) → 𝑅 ∈ (TopOn‘𝑋))
3 topontop 20997 . . . . . 6 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
42, 3syl 17 . . . . 5 ((𝜑𝑘𝐴) → 𝑅 ∈ Top)
5 ptcls.c . . . . . . 7 ((𝜑𝑘𝐴) → 𝑆𝑋)
6 toponuni 20998 . . . . . . . 8 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
72, 6syl 17 . . . . . . 7 ((𝜑𝑘𝐴) → 𝑋 = 𝑅)
85, 7sseqtrd 3801 . . . . . 6 ((𝜑𝑘𝐴) → 𝑆 𝑅)
9 eqid 2765 . . . . . . 7 𝑅 = 𝑅
109clscld 21131 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
114, 8, 10syl2anc 579 . . . . 5 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
121, 4, 11ptcldmpt 21697 . . . 4 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘(∏t‘(𝑘𝐴𝑅))))
13 ptcls.2 . . . . 5 𝐽 = (∏t‘(𝑘𝐴𝑅))
1413fveq2i 6378 . . . 4 (Clsd‘𝐽) = (Clsd‘(∏t‘(𝑘𝐴𝑅)))
1512, 14syl6eleqr 2855 . . 3 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽))
169sscls 21140 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
174, 8, 16syl2anc 579 . . . . 5 ((𝜑𝑘𝐴) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
1817ralrimiva 3113 . . . 4 (𝜑 → ∀𝑘𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
19 ss2ixp 8126 . . . 4 (∀𝑘𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆) → X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆))
2018, 19syl 17 . . 3 (𝜑X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆))
21 eqid 2765 . . . 4 𝐽 = 𝐽
2221clsss2 21156 . . 3 ((X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽) ∧ X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → ((cls‘𝐽)‘X𝑘𝐴 𝑆) ⊆ X𝑘𝐴 ((cls‘𝑅)‘𝑆))
2315, 20, 22syl2anc 579 . 2 (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) ⊆ X𝑘𝐴 ((cls‘𝑅)‘𝑆))
24 vex 3353 . . . . . 6 𝑢 ∈ V
25 eqeq1 2769 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑢 = X𝑦𝐴 (𝑔𝑦)))
2625anbi2d 622 . . . . . . 7 (𝑥 = 𝑢 → (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) ↔ ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑢 = X𝑦𝐴 (𝑔𝑦))))
2726exbidv 2016 . . . . . 6 (𝑥 = 𝑢 → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑢 = X𝑦𝐴 (𝑔𝑦))))
2824, 27elab 3504 . . . . 5 (𝑢 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑢 = X𝑦𝐴 (𝑔𝑦)))
29 nffvmpt1 6386 . . . . . . . . . . . . . . . 16 𝑘((𝑘𝐴𝑅)‘𝑦)
3029nfel2 2924 . . . . . . . . . . . . . . 15 𝑘(𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)
31 nfv 2009 . . . . . . . . . . . . . . 15 𝑦(𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘)
32 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑔𝑦) = (𝑔𝑘))
33 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → ((𝑘𝐴𝑅)‘𝑦) = ((𝑘𝐴𝑅)‘𝑘))
3432, 33eleq12d 2838 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → ((𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘)))
3530, 31, 34cbvral 3315 . . . . . . . . . . . . . 14 (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘))
36 simpr 477 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → 𝑘𝐴)
37 eqid 2765 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴𝑅) = (𝑘𝐴𝑅)
3837fvmpt2 6480 . . . . . . . . . . . . . . . . 17 ((𝑘𝐴𝑅 ∈ (TopOn‘𝑋)) → ((𝑘𝐴𝑅)‘𝑘) = 𝑅)
3936, 2, 38syl2anc 579 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑅)‘𝑘) = 𝑅)
4039eleq2d 2830 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → ((𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘) ↔ (𝑔𝑘) ∈ 𝑅))
4140ralbidva 3132 . . . . . . . . . . . . . 14 (𝜑 → (∀𝑘𝐴 (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅))
4235, 41syl5bb 274 . . . . . . . . . . . . 13 (𝜑 → (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅))
4342anbi2d 622 . . . . . . . . . . . 12 (𝜑 → ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)))
4443adantr 472 . . . . . . . . . . 11 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)))
4544biimpa 468 . . . . . . . . . 10 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦))) → (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅))
46 ptclsg.1 . . . . . . . . . . . . . 14 (𝜑 𝑘𝐴 𝑆AC 𝐴)
4746ad2antrr 717 . . . . . . . . . . . . 13 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑘𝐴 𝑆AC 𝐴)
48 simpll 783 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝜑)
49 vex 3353 . . . . . . . . . . . . . . . . . . . 20 𝑓 ∈ V
5049elixp 8120 . . . . . . . . . . . . . . . . . . 19 (𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)))
5150simprbi 490 . . . . . . . . . . . . . . . . . 18 (𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆) → ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆))
5251ad2antlr 718 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆))
539clsndisj 21159 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 𝑅 ∧ (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)) ∧ ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘))) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
5453ex 401 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Top ∧ 𝑆 𝑅 ∧ (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
55543expia 1150 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)))
564, 8, 55syl2anc 579 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → ((𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)))
5756ralimdva 3109 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆) → ∀𝑘𝐴 (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)))
5848, 52, 57sylc 65 . . . . . . . . . . . . . . . 16 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
59 simprlr 798 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)
60 simprr 789 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑓X𝑦𝐴 (𝑔𝑦))
6132cbvixpv 8131 . . . . . . . . . . . . . . . . . . 19 X𝑦𝐴 (𝑔𝑦) = X𝑘𝐴 (𝑔𝑘)
6260, 61syl6eleq 2854 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑓X𝑘𝐴 (𝑔𝑘))
6349elixp 8120 . . . . . . . . . . . . . . . . . . 19 (𝑓X𝑘𝐴 (𝑔𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘)))
6463simprbi 490 . . . . . . . . . . . . . . . . . 18 (𝑓X𝑘𝐴 (𝑔𝑘) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘))
6562, 64syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘))
66 r19.26 3211 . . . . . . . . . . . . . . . . 17 (∀𝑘𝐴 ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) ↔ (∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘)))
6759, 65, 66sylanbrc 578 . . . . . . . . . . . . . . . 16 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)))
68 ralim 3095 . . . . . . . . . . . . . . . 16 (∀𝑘𝐴 (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅) → (∀𝑘𝐴 ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ∀𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
6958, 67, 68sylc 65 . . . . . . . . . . . . . . 15 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
70 rabn0 4122 . . . . . . . . . . . . . . . . 17 ({𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} ≠ ∅ ↔ ∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆))
71 dfin5 3740 . . . . . . . . . . . . . . . . . . 19 ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = {𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)}
72 inss2 3993 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔𝑘) ∩ 𝑆) ⊆ 𝑆
73 ssiun2 4719 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝐴𝑆 𝑘𝐴 𝑆)
7472, 73syl5ss 3772 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐴 → ((𝑔𝑘) ∩ 𝑆) ⊆ 𝑘𝐴 𝑆)
75 sseqin2 3979 . . . . . . . . . . . . . . . . . . . 20 (((𝑔𝑘) ∩ 𝑆) ⊆ 𝑘𝐴 𝑆 ↔ ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = ((𝑔𝑘) ∩ 𝑆))
7674, 75sylib 209 . . . . . . . . . . . . . . . . . . 19 (𝑘𝐴 → ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = ((𝑔𝑘) ∩ 𝑆))
7771, 76syl5eqr 2813 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴 → {𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} = ((𝑔𝑘) ∩ 𝑆))
7877neeq1d 2996 . . . . . . . . . . . . . . . . 17 (𝑘𝐴 → ({𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} ≠ ∅ ↔ ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
7970, 78syl5bbr 276 . . . . . . . . . . . . . . . 16 (𝑘𝐴 → (∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
8079ralbiia 3126 . . . . . . . . . . . . . . 15 (∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
8169, 80sylibr 225 . . . . . . . . . . . . . 14 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆))
82 nfv 2009 . . . . . . . . . . . . . . 15 𝑦𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)
83 nfiu1 4706 . . . . . . . . . . . . . . . 16 𝑘 𝑘𝐴 𝑆
84 nfcv 2907 . . . . . . . . . . . . . . . . . 18 𝑘(𝑔𝑦)
85 nfcsb1v 3707 . . . . . . . . . . . . . . . . . 18 𝑘𝑦 / 𝑘𝑆
8684, 85nfin 3980 . . . . . . . . . . . . . . . . 17 𝑘((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
8786nfel2 2924 . . . . . . . . . . . . . . . 16 𝑘 𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
8883, 87nfrex 3153 . . . . . . . . . . . . . . 15 𝑘𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
89 fveq2 6375 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦 → (𝑔𝑘) = (𝑔𝑦))
90 csbeq1a 3700 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦𝑆 = 𝑦 / 𝑘𝑆)
9189, 90ineq12d 3977 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → ((𝑔𝑘) ∩ 𝑆) = ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
9291eleq2d 2830 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → (𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ 𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9392rexbidv 3199 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9482, 88, 93cbvral 3315 . . . . . . . . . . . . . 14 (∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
9581, 94sylib 209 . . . . . . . . . . . . 13 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
96 eleq1 2832 . . . . . . . . . . . . . 14 (𝑧 = (𝑦) → (𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆) ↔ (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9796acni3 9121 . . . . . . . . . . . . 13 (( 𝑘𝐴 𝑆AC 𝐴 ∧ ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → ∃(:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9847, 95, 97syl2anc 579 . . . . . . . . . . . 12 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∃(:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
99 ffn 6223 . . . . . . . . . . . . . 14 (:𝐴 𝑘𝐴 𝑆 Fn 𝐴)
100 nfv 2009 . . . . . . . . . . . . . . . 16 𝑦(𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)
10186nfel2 2924 . . . . . . . . . . . . . . . 16 𝑘(𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
102 fveq2 6375 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → (𝑘) = (𝑦))
103102, 91eleq12d 2838 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → ((𝑘) ∈ ((𝑔𝑘) ∩ 𝑆) ↔ (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
104100, 101, 103cbvral 3315 . . . . . . . . . . . . . . 15 (∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
105 ne0i 4085 . . . . . . . . . . . . . . . 16 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) → X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
106 vex 3353 . . . . . . . . . . . . . . . . 17 ∈ V
107106elixp 8120 . . . . . . . . . . . . . . . 16 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ↔ ( Fn 𝐴 ∧ ∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)))
108 ixpin 8138 . . . . . . . . . . . . . . . . . 18 X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) = (X𝑘𝐴 (𝑔𝑘) ∩ X𝑘𝐴 𝑆)
10961ineq1i 3972 . . . . . . . . . . . . . . . . . 18 (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) = (X𝑘𝐴 (𝑔𝑘) ∩ X𝑘𝐴 𝑆)
110108, 109eqtr4i 2790 . . . . . . . . . . . . . . . . 17 X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) = (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆)
111110neeq1i 3001 . . . . . . . . . . . . . . . 16 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅ ↔ (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
112105, 107, 1113imtr3i 282 . . . . . . . . . . . . . . 15 (( Fn 𝐴 ∧ ∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
113104, 112sylan2br 588 . . . . . . . . . . . . . 14 (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
11499, 113sylan 575 . . . . . . . . . . . . 13 ((:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
115114exlimiv 2025 . . . . . . . . . . . 12 (∃(:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
11698, 115syl 17 . . . . . . . . . . 11 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
117116expr 448 . . . . . . . . . 10 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)) → (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
11845, 117syldan 585 . . . . . . . . 9 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦))) → (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
1191183adantr3 1212 . . . . . . . 8 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦))) → (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
120 eleq2 2833 . . . . . . . . 9 (𝑢 = X𝑦𝐴 (𝑔𝑦) → (𝑓𝑢𝑓X𝑦𝐴 (𝑔𝑦)))
121 ineq1 3969 . . . . . . . . . 10 (𝑢 = X𝑦𝐴 (𝑔𝑦) → (𝑢X𝑘𝐴 𝑆) = (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆))
122121neeq1d 2996 . . . . . . . . 9 (𝑢 = X𝑦𝐴 (𝑔𝑦) → ((𝑢X𝑘𝐴 𝑆) ≠ ∅ ↔ (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
123120, 122imbi12d 335 . . . . . . . 8 (𝑢 = X𝑦𝐴 (𝑔𝑦) → ((𝑓𝑢 → (𝑢X𝑘𝐴 𝑆) ≠ ∅) ↔ (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)))
124119, 123syl5ibrcom 238 . . . . . . 7 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦))) → (𝑢 = X𝑦𝐴 (𝑔𝑦) → (𝑓𝑢 → (𝑢X𝑘𝐴 𝑆) ≠ ∅)))
125124expimpd 445 . . . . . 6 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑢 = X𝑦𝐴 (𝑔𝑦)) → (𝑓𝑢 → (𝑢X𝑘𝐴 𝑆) ≠ ∅)))
126125exlimdv 2028 . . . . 5 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑢 = X𝑦𝐴 (𝑔𝑦)) → (𝑓𝑢 → (𝑢X𝑘𝐴 𝑆) ≠ ∅)))
12728, 126syl5bi 233 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → (𝑢 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} → (𝑓𝑢 → (𝑢X𝑘𝐴 𝑆) ≠ ∅)))
128127ralrimiv 3112 . . 3 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → ∀𝑢 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} (𝑓𝑢 → (𝑢X𝑘𝐴 𝑆) ≠ ∅))
1294fmpttd 6575 . . . . . . . 8 (𝜑 → (𝑘𝐴𝑅):𝐴⟶Top)
130129ffnd 6224 . . . . . . 7 (𝜑 → (𝑘𝐴𝑅) Fn 𝐴)
131 eqid 2765 . . . . . . . 8 {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
132131ptval 21653 . . . . . . 7 ((𝐴𝑉 ∧ (𝑘𝐴𝑅) Fn 𝐴) → (∏t‘(𝑘𝐴𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
1331, 130, 132syl2anc 579 . . . . . 6 (𝜑 → (∏t‘(𝑘𝐴𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
13413, 133syl5eq 2811 . . . . 5 (𝜑𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
135134adantr 472 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → 𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
1362ralrimiva 3113 . . . . . . 7 (𝜑 → ∀𝑘𝐴 𝑅 ∈ (TopOn‘𝑋))
13713pttopon 21679 . . . . . . 7 ((𝐴𝑉 ∧ ∀𝑘𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋))
1381, 136, 137syl2anc 579 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋))
139 toponuni 20998 . . . . . 6 (𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋) → X𝑘𝐴 𝑋 = 𝐽)
140138, 139syl 17 . . . . 5 (𝜑X𝑘𝐴 𝑋 = 𝐽)
141140adantr 472 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘𝐴 𝑋 = 𝐽)
142131ptbas 21662 . . . . . 6 ((𝐴𝑉 ∧ (𝑘𝐴𝑅):𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
1431, 129, 142syl2anc 579 . . . . 5 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
144143adantr 472 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
1455ralrimiva 3113 . . . . . 6 (𝜑 → ∀𝑘𝐴 𝑆𝑋)
146 ss2ixp 8126 . . . . . 6 (∀𝑘𝐴 𝑆𝑋X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
147145, 146syl 17 . . . . 5 (𝜑X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
148147adantr 472 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
1499clsss3 21143 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((cls‘𝑅)‘𝑆) ⊆ 𝑅)
1504, 8, 149syl2anc 579 . . . . . . . 8 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ⊆ 𝑅)
151150, 7sseqtr4d 3802 . . . . . . 7 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
152151ralrimiva 3113 . . . . . 6 (𝜑 → ∀𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
153 ss2ixp 8126 . . . . . 6 (∀𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋X𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ X𝑘𝐴 𝑋)
154152, 153syl 17 . . . . 5 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ X𝑘𝐴 𝑋)
155154sselda 3761 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓X𝑘𝐴 𝑋)
156135, 141, 144, 148, 155elcls3 21167 . . 3 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → (𝑓 ∈ ((cls‘𝐽)‘X𝑘𝐴 𝑆) ↔ ∀𝑢 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} (𝑓𝑢 → (𝑢X𝑘𝐴 𝑆) ≠ ∅)))
157128, 156mpbird 248 . 2 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓 ∈ ((cls‘𝐽)‘X𝑘𝐴 𝑆))
15823, 157eqelssd 3782 1 (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) = X𝑘𝐴 ((cls‘𝑅)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  csb 3691  cdif 3729  cin 3731  wss 3732  c0 4079   cuni 4594   ciun 4676  cmpt 4888   Fn wfn 6063  wf 6064  cfv 6068  Xcixp 8113  Fincfn 8160  AC wacn 9015  topGenctg 16366  tcpt 16367  Topctop 20977  TopOnctopon 20994  TopBasesctb 21029  Clsdccld 21100  clsccl 21102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-ixp 8114  df-en 8161  df-fin 8164  df-fi 8524  df-acn 9019  df-topgen 16372  df-pt 16373  df-top 20978  df-topon 20995  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105
This theorem is referenced by:  ptcls  21699  dfac14  21701
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