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Theorem ptclsg 22329
 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 21627 . . . . . 6 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
42, 3syl 17 . . . . 5 ((𝜑𝑘𝐴) → 𝑅 ∈ Top)
5 ptcls.c . . . . . . 7 ((𝜑𝑘𝐴) → 𝑆𝑋)
6 toponuni 21628 . . . . . . . 8 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
72, 6syl 17 . . . . . . 7 ((𝜑𝑘𝐴) → 𝑋 = 𝑅)
85, 7sseqtrd 3934 . . . . . 6 ((𝜑𝑘𝐴) → 𝑆 𝑅)
9 eqid 2758 . . . . . . 7 𝑅 = 𝑅
109clscld 21761 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
114, 8, 10syl2anc 587 . . . . 5 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
121, 4, 11ptcldmpt 22328 . . . 4 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘(∏t‘(𝑘𝐴𝑅))))
13 ptcls.2 . . . . 5 𝐽 = (∏t‘(𝑘𝐴𝑅))
1413fveq2i 6666 . . . 4 (Clsd‘𝐽) = (Clsd‘(∏t‘(𝑘𝐴𝑅)))
1512, 14eleqtrrdi 2863 . . 3 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽))
169sscls 21770 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
174, 8, 16syl2anc 587 . . . . 5 ((𝜑𝑘𝐴) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
1817ralrimiva 3113 . . . 4 (𝜑 → ∀𝑘𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
19 ss2ixp 8505 . . . 4 (∀𝑘𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆) → X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆))
2018, 19syl 17 . . 3 (𝜑X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆))
21 eqid 2758 . . . 4 𝐽 = 𝐽
2221clsss2 21786 . . 3 ((X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽) ∧ X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → ((cls‘𝐽)‘X𝑘𝐴 𝑆) ⊆ X𝑘𝐴 ((cls‘𝑅)‘𝑆))
2315, 20, 22syl2anc 587 . 2 (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) ⊆ X𝑘𝐴 ((cls‘𝑅)‘𝑆))
24 vex 3413 . . . . . 6 𝑢 ∈ V
25 eqeq1 2762 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑢 = X𝑦𝐴 (𝑔𝑦)))
2625anbi2d 631 . . . . . . 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 1922 . . . . . 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 3590 . . . . 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 6674 . . . . . . . . . . . . . . . 16 𝑘((𝑘𝐴𝑅)‘𝑦)
3029nfel2 2937 . . . . . . . . . . . . . . 15 𝑘(𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)
31 nfv 1915 . . . . . . . . . . . . . . 15 𝑦(𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘)
32 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑔𝑦) = (𝑔𝑘))
33 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → ((𝑘𝐴𝑅)‘𝑦) = ((𝑘𝐴𝑅)‘𝑘))
3432, 33eleq12d 2846 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → ((𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘)))
3530, 31, 34cbvralw 3352 . . . . . . . . . . . . . 14 (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘))
36 simpr 488 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → 𝑘𝐴)
37 eqid 2758 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴𝑅) = (𝑘𝐴𝑅)
3837fvmpt2 6775 . . . . . . . . . . . . . . . . 17 ((𝑘𝐴𝑅 ∈ (TopOn‘𝑋)) → ((𝑘𝐴𝑅)‘𝑘) = 𝑅)
3936, 2, 38syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑅)‘𝑘) = 𝑅)
4039eleq2d 2837 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → ((𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘) ↔ (𝑔𝑘) ∈ 𝑅))
4140ralbidva 3125 . . . . . . . . . . . . . 14 (𝜑 → (∀𝑘𝐴 (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅))
4235, 41syl5bb 286 . . . . . . . . . . . . 13 (𝜑 → (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅))
4342anbi2d 631 . . . . . . . . . . . 12 (𝜑 → ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)))
4443adantr 484 . . . . . . . . . . 11 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)))
4544biimpa 480 . . . . . . . . . 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 725 . . . . . . . . . . . . 13 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑘𝐴 𝑆AC 𝐴)
48 simpll 766 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝜑)
49 vex 3413 . . . . . . . . . . . . . . . . . . . 20 𝑓 ∈ V
5049elixp 8499 . . . . . . . . . . . . . . . . . . 19 (𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)))
5150simprbi 500 . . . . . . . . . . . . . . . . . 18 (𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆) → ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆))
5251ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆))
539clsndisj 21789 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 𝑅 ∧ (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)) ∧ ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘))) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
5453ex 416 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Top ∧ 𝑆 𝑅 ∧ (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
55543expia 1118 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)))
564, 8, 55syl2anc 587 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → ((𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)))
5756ralimdva 3108 . . . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)
60 simprr 772 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑓X𝑦𝐴 (𝑔𝑦))
6132cbvixpv 8510 . . . . . . . . . . . . . . . . . . 19 X𝑦𝐴 (𝑔𝑦) = X𝑘𝐴 (𝑔𝑘)
6260, 61eleqtrdi 2862 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑓X𝑘𝐴 (𝑔𝑘))
6349elixp 8499 . . . . . . . . . . . . . . . . . . 19 (𝑓X𝑘𝐴 (𝑔𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘)))
6463simprbi 500 . . . . . . . . . . . . . . . . . 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 3101 . . . . . . . . . . . . . . . . 17 (∀𝑘𝐴 ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) ↔ (∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘)))
6759, 65, 66sylanbrc 586 . . . . . . . . . . . . . . . 16 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)))
68 ralim 3094 . . . . . . . . . . . . . . . 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 4284 . . . . . . . . . . . . . . . . 17 ({𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} ≠ ∅ ↔ ∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆))
71 dfin5 3868 . . . . . . . . . . . . . . . . . . 19 ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = {𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)}
72 inss2 4136 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔𝑘) ∩ 𝑆) ⊆ 𝑆
73 ssiun2 4939 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝐴𝑆 𝑘𝐴 𝑆)
7472, 73sstrid 3905 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐴 → ((𝑔𝑘) ∩ 𝑆) ⊆ 𝑘𝐴 𝑆)
75 sseqin2 4122 . . . . . . . . . . . . . . . . . . . 20 (((𝑔𝑘) ∩ 𝑆) ⊆ 𝑘𝐴 𝑆 ↔ ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = ((𝑔𝑘) ∩ 𝑆))
7674, 75sylib 221 . . . . . . . . . . . . . . . . . . 19 (𝑘𝐴 → ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = ((𝑔𝑘) ∩ 𝑆))
7771, 76eqtr3id 2807 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴 → {𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} = ((𝑔𝑘) ∩ 𝑆))
7877neeq1d 3010 . . . . . . . . . . . . . . . . 17 (𝑘𝐴 → ({𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} ≠ ∅ ↔ ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
7970, 78bitr3id 288 . . . . . . . . . . . . . . . 16 (𝑘𝐴 → (∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
8079ralbiia 3096 . . . . . . . . . . . . . . 15 (∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
8169, 80sylibr 237 . . . . . . . . . . . . . 14 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆))
82 nfv 1915 . . . . . . . . . . . . . . 15 𝑦𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)
83 nfiu1 4920 . . . . . . . . . . . . . . . 16 𝑘 𝑘𝐴 𝑆
84 nfcv 2919 . . . . . . . . . . . . . . . . . 18 𝑘(𝑔𝑦)
85 nfcsb1v 3831 . . . . . . . . . . . . . . . . . 18 𝑘𝑦 / 𝑘𝑆
8684, 85nfin 4123 . . . . . . . . . . . . . . . . 17 𝑘((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
8786nfel2 2937 . . . . . . . . . . . . . . . 16 𝑘 𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
8883, 87nfrex 3233 . . . . . . . . . . . . . . 15 𝑘𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
89 fveq2 6663 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦 → (𝑔𝑘) = (𝑔𝑦))
90 csbeq1a 3821 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦𝑆 = 𝑦 / 𝑘𝑆)
9189, 90ineq12d 4120 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → ((𝑔𝑘) ∩ 𝑆) = ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
9291eleq2d 2837 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → (𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ 𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9392rexbidv 3221 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9482, 88, 93cbvralw 3352 . . . . . . . . . . . . . 14 (∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
9581, 94sylib 221 . . . . . . . . . . . . 13 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
96 eleq1 2839 . . . . . . . . . . . . . 14 (𝑧 = (𝑦) → (𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆) ↔ (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9796acni3 9520 . . . . . . . . . . . . 13 (( 𝑘𝐴 𝑆AC 𝐴 ∧ ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → ∃(:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9847, 95, 97syl2anc 587 . . . . . . . . . . . 12 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∃(:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
99 ffn 6503 . . . . . . . . . . . . . 14 (:𝐴 𝑘𝐴 𝑆 Fn 𝐴)
100 nfv 1915 . . . . . . . . . . . . . . . 16 𝑦(𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)
10186nfel2 2937 . . . . . . . . . . . . . . . 16 𝑘(𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
102 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → (𝑘) = (𝑦))
103102, 91eleq12d 2846 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → ((𝑘) ∈ ((𝑔𝑘) ∩ 𝑆) ↔ (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
104100, 101, 103cbvralw 3352 . . . . . . . . . . . . . . 15 (∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
105 ne0i 4235 . . . . . . . . . . . . . . . 16 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) → X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
106 vex 3413 . . . . . . . . . . . . . . . . 17 ∈ V
107106elixp 8499 . . . . . . . . . . . . . . . 16 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ↔ ( Fn 𝐴 ∧ ∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)))
108 ixpin 8518 . . . . . . . . . . . . . . . . . 18 X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) = (X𝑘𝐴 (𝑔𝑘) ∩ X𝑘𝐴 𝑆)
10961ineq1i 4115 . . . . . . . . . . . . . . . . . 18 (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) = (X𝑘𝐴 (𝑔𝑘) ∩ X𝑘𝐴 𝑆)
110108, 109eqtr4i 2784 . . . . . . . . . . . . . . . . 17 X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) = (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆)
111110neeq1i 3015 . . . . . . . . . . . . . . . 16 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅ ↔ (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
112105, 107, 1113imtr3i 294 . . . . . . . . . . . . . . 15 (( Fn 𝐴 ∧ ∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
113104, 112sylan2br 597 . . . . . . . . . . . . . 14 (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
11499, 113sylan 583 . . . . . . . . . . . . 13 ((:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
115114exlimiv 1931 . . . . . . . . . . . 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 460 . . . . . . . . . 10 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)) → (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
11845, 117syldan 594 . . . . . . . . 9 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦))) → (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
1191183adantr3 1168 . . . . . . . 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 2840 . . . . . . . . 9 (𝑢 = X𝑦𝐴 (𝑔𝑦) → (𝑓𝑢𝑓X𝑦𝐴 (𝑔𝑦)))
121 ineq1 4111 . . . . . . . . . 10 (𝑢 = X𝑦𝐴 (𝑔𝑦) → (𝑢X𝑘𝐴 𝑆) = (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆))
122121neeq1d 3010 . . . . . . . . 9 (𝑢 = X𝑦𝐴 (𝑔𝑦) → ((𝑢X𝑘𝐴 𝑆) ≠ ∅ ↔ (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
123120, 122imbi12d 348 . . . . . . . 8 (𝑢 = X𝑦𝐴 (𝑔𝑦) → ((𝑓𝑢 → (𝑢X𝑘𝐴 𝑆) ≠ ∅) ↔ (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)))
124119, 123syl5ibrcom 250 . . . . . . 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 457 . . . . . 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 1934 . . . . 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 245 . . . 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 6876 . . . . . . . 8 (𝜑 → (𝑘𝐴𝑅):𝐴⟶Top)
130129ffnd 6504 . . . . . . 7 (𝜑 → (𝑘𝐴𝑅) Fn 𝐴)
131 eqid 2758 . . . . . . . 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 22284 . . . . . . 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 587 . . . . . 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 2805 . . . . 5 (𝜑𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
135134adantr 484 . . . 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 22310 . . . . . . 7 ((𝐴𝑉 ∧ ∀𝑘𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋))
1381, 136, 137syl2anc 587 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋))
139 toponuni 21628 . . . . . 6 (𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋) → X𝑘𝐴 𝑋 = 𝐽)
140138, 139syl 17 . . . . 5 (𝜑X𝑘𝐴 𝑋 = 𝐽)
141140adantr 484 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘𝐴 𝑋 = 𝐽)
142131ptbas 22293 . . . . . 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 587 . . . . 5 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
144143adantr 484 . . . 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 8505 . . . . . 6 (∀𝑘𝐴 𝑆𝑋X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
147145, 146syl 17 . . . . 5 (𝜑X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
148147adantr 484 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
1499clsss3 21773 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((cls‘𝑅)‘𝑆) ⊆ 𝑅)
1504, 8, 149syl2anc 587 . . . . . . . 8 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ⊆ 𝑅)
151150, 7sseqtrrd 3935 . . . . . . 7 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
152151ralrimiva 3113 . . . . . 6 (𝜑 → ∀𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
153 ss2ixp 8505 . . . . . 6 (∀𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋X𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ X𝑘𝐴 𝑋)
154152, 153syl 17 . . . . 5 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ X𝑘𝐴 𝑋)
155154sselda 3894 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓X𝑘𝐴 𝑋)
156135, 141, 144, 148, 155elcls3 21797 . . 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 260 . 2 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓 ∈ ((cls‘𝐽)‘X𝑘𝐴 𝑆))
15823, 157eqelssd 3915 1 (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) = X𝑘𝐴 ((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 2735   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074  ⦋csb 3807   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  ∪ cuni 4801  ∪ ciun 4886   ↦ cmpt 5116   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  Xcixp 8492  Fincfn 8540  AC wacn 9413  topGenctg 16783  ∏tcpt 16784  Topctop 21607  TopOnctopon 21624  TopBasesctb 21659  Clsdccld 21730  clsccl 21732 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 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1o 8118  df-er 8305  df-map 8424  df-ixp 8493  df-en 8541  df-fin 8544  df-fi 8921  df-acn 9417  df-topgen 16789  df-pt 16790  df-top 21608  df-topon 21625  df-bases 21660  df-cld 21733  df-ntr 21734  df-cls 21735 This theorem is referenced by:  ptcls  22330  dfac14  22332
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