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Theorem ptclsg 21337
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 20646 . . . . . 6 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
42, 3syl 17 . . . . 5 ((𝜑𝑘𝐴) → 𝑅 ∈ Top)
5 ptcls.c . . . . . . 7 ((𝜑𝑘𝐴) → 𝑆𝑋)
6 toponuni 20647 . . . . . . . 8 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
72, 6syl 17 . . . . . . 7 ((𝜑𝑘𝐴) → 𝑋 = 𝑅)
85, 7sseqtrd 3625 . . . . . 6 ((𝜑𝑘𝐴) → 𝑆 𝑅)
9 eqid 2621 . . . . . . 7 𝑅 = 𝑅
109clscld 20770 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
114, 8, 10syl2anc 692 . . . . 5 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
121, 4, 11ptcldmpt 21336 . . . 4 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘(∏t‘(𝑘𝐴𝑅))))
13 ptcls.2 . . . . 5 𝐽 = (∏t‘(𝑘𝐴𝑅))
1413fveq2i 6156 . . . 4 (Clsd‘𝐽) = (Clsd‘(∏t‘(𝑘𝐴𝑅)))
1512, 14syl6eleqr 2709 . . 3 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽))
169sscls 20779 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
174, 8, 16syl2anc 692 . . . . 5 ((𝜑𝑘𝐴) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
1817ralrimiva 2961 . . . 4 (𝜑 → ∀𝑘𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
19 ss2ixp 7872 . . . 4 (∀𝑘𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆) → X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆))
2018, 19syl 17 . . 3 (𝜑X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆))
21 eqid 2621 . . . 4 𝐽 = 𝐽
2221clsss2 20795 . . 3 ((X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽) ∧ X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → ((cls‘𝐽)‘X𝑘𝐴 𝑆) ⊆ X𝑘𝐴 ((cls‘𝑅)‘𝑆))
2315, 20, 22syl2anc 692 . 2 (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) ⊆ X𝑘𝐴 ((cls‘𝑅)‘𝑆))
24 vex 3192 . . . . . . . 8 𝑢 ∈ V
25 eqeq1 2625 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑥 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑢 = X𝑦𝐴 (𝑔𝑦)))
2625anbi2d 739 . . . . . . . . 9 (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 1847 . . . . . . . 8 (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 3337 . . . . . . 7 (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 6161 . . . . . . . . . . . . . . . . . 18 𝑘((𝑘𝐴𝑅)‘𝑦)
3029nfel2 2777 . . . . . . . . . . . . . . . . 17 𝑘(𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)
31 nfv 1840 . . . . . . . . . . . . . . . . 17 𝑦(𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘)
32 fveq2 6153 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑘 → (𝑔𝑦) = (𝑔𝑘))
33 fveq2 6153 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑘 → ((𝑘𝐴𝑅)‘𝑦) = ((𝑘𝐴𝑅)‘𝑘))
3432, 33eleq12d 2692 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑘 → ((𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘)))
3530, 31, 34cbvral 3158 . . . . . . . . . . . . . . . 16 (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘))
36 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐴) → 𝑘𝐴)
37 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐴𝑅) = (𝑘𝐴𝑅)
3837fvmpt2 6253 . . . . . . . . . . . . . . . . . . 19 ((𝑘𝐴𝑅 ∈ (TopOn‘𝑋)) → ((𝑘𝐴𝑅)‘𝑘) = 𝑅)
3936, 2, 38syl2anc 692 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑅)‘𝑘) = 𝑅)
4039eleq2d 2684 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → ((𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘) ↔ (𝑔𝑘) ∈ 𝑅))
4140ralbidva 2980 . . . . . . . . . . . . . . . 16 (𝜑 → (∀𝑘𝐴 (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅))
4235, 41syl5bb 272 . . . . . . . . . . . . . . 15 (𝜑 → (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅))
4342anbi2d 739 . . . . . . . . . . . . . 14 (𝜑 → ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)))
4443adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)))
4544biimpa 501 . . . . . . . . . . . 12 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦))) → (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅))
46 ptclsg.1 . . . . . . . . . . . . . . . 16 (𝜑 𝑘𝐴 𝑆AC 𝐴)
4746ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑘𝐴 𝑆AC 𝐴)
48 simpll 789 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝜑)
49 vex 3192 . . . . . . . . . . . . . . . . . . . . . 22 𝑓 ∈ V
5049elixp 7866 . . . . . . . . . . . . . . . . . . . . 21 (𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)))
5150simprbi 480 . . . . . . . . . . . . . . . . . . . 20 (𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆) → ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆))
5251ad2antlr 762 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆))
539clsndisj 20798 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Top ∧ 𝑆 𝑅 ∧ (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)) ∧ ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘))) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
5453ex 450 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Top ∧ 𝑆 𝑅 ∧ (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
55543expia 1264 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)))
564, 8, 55syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐴) → ((𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)))
5756ralimdva 2957 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆) → ∀𝑘𝐴 (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)))
5848, 52, 57sylc 65 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
59 simprlr 802 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)
60 simprr 795 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑓X𝑦𝐴 (𝑔𝑦))
6132cbvixpv 7877 . . . . . . . . . . . . . . . . . . . . 21 X𝑦𝐴 (𝑔𝑦) = X𝑘𝐴 (𝑔𝑘)
6260, 61syl6eleq 2708 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑓X𝑘𝐴 (𝑔𝑘))
6349elixp 7866 . . . . . . . . . . . . . . . . . . . . 21 (𝑓X𝑘𝐴 (𝑔𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘)))
6463simprbi 480 . . . . . . . . . . . . . . . . . . . 20 (𝑓X𝑘𝐴 (𝑔𝑘) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘))
6562, 64syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘))
66 r19.26 3058 . . . . . . . . . . . . . . . . . . 19 (∀𝑘𝐴 ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) ↔ (∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘)))
6759, 65, 66sylanbrc 697 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)))
68 ralim 2943 . . . . . . . . . . . . . . . . . 18 (∀𝑘𝐴 (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅) → (∀𝑘𝐴 ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ∀𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
6958, 67, 68sylc 65 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
70 rabn0 3937 . . . . . . . . . . . . . . . . . . 19 ({𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} ≠ ∅ ↔ ∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆))
71 dfin5 3567 . . . . . . . . . . . . . . . . . . . . 21 ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = {𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)}
72 inss2 3817 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔𝑘) ∩ 𝑆) ⊆ 𝑆
73 ssiun2 4534 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘𝐴𝑆 𝑘𝐴 𝑆)
7472, 73syl5ss 3598 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝐴 → ((𝑔𝑘) ∩ 𝑆) ⊆ 𝑘𝐴 𝑆)
75 sseqin2 3800 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔𝑘) ∩ 𝑆) ⊆ 𝑘𝐴 𝑆 ↔ ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = ((𝑔𝑘) ∩ 𝑆))
7674, 75sylib 208 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝐴 → ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = ((𝑔𝑘) ∩ 𝑆))
7771, 76syl5eqr 2669 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐴 → {𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} = ((𝑔𝑘) ∩ 𝑆))
7877neeq1d 2849 . . . . . . . . . . . . . . . . . . 19 (𝑘𝐴 → ({𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} ≠ ∅ ↔ ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
7970, 78syl5bbr 274 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴 → (∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
8079ralbiia 2974 . . . . . . . . . . . . . . . . 17 (∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
8169, 80sylibr 224 . . . . . . . . . . . . . . . 16 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆))
82 nfv 1840 . . . . . . . . . . . . . . . . 17 𝑦𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)
83 nfiu1 4521 . . . . . . . . . . . . . . . . . 18 𝑘 𝑘𝐴 𝑆
84 nfcv 2761 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝑔𝑦)
85 nfcsb1v 3534 . . . . . . . . . . . . . . . . . . . 20 𝑘𝑦 / 𝑘𝑆
8684, 85nfin 3803 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
8786nfel2 2777 . . . . . . . . . . . . . . . . . 18 𝑘 𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
8883, 87nfrex 3002 . . . . . . . . . . . . . . . . 17 𝑘𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
89 fveq2 6153 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑦 → (𝑔𝑘) = (𝑔𝑦))
90 csbeq1a 3527 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑦𝑆 = 𝑦 / 𝑘𝑆)
9189, 90ineq12d 3798 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑦 → ((𝑔𝑘) ∩ 𝑆) = ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
9291eleq2d 2684 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦 → (𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ 𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9392rexbidv 3046 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → (∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9482, 88, 93cbvral 3158 . . . . . . . . . . . . . . . 16 (∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
9581, 94sylib 208 . . . . . . . . . . . . . . 15 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
96 eleq1 2686 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑦) → (𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆) ↔ (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9796acni3 8821 . . . . . . . . . . . . . . 15 (( 𝑘𝐴 𝑆AC 𝐴 ∧ ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → ∃(:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9847, 95, 97syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∃(:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
99 ffn 6007 . . . . . . . . . . . . . . . 16 (:𝐴 𝑘𝐴 𝑆 Fn 𝐴)
100 nfv 1840 . . . . . . . . . . . . . . . . . 18 𝑦(𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)
10186nfel2 2777 . . . . . . . . . . . . . . . . . 18 𝑘(𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
102 fveq2 6153 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑦 → (𝑘) = (𝑦))
103102, 91eleq12d 2692 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦 → ((𝑘) ∈ ((𝑔𝑘) ∩ 𝑆) ↔ (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
104100, 101, 103cbvral 3158 . . . . . . . . . . . . . . . . 17 (∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
105 ne0i 3902 . . . . . . . . . . . . . . . . . 18 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) → X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
106 vex 3192 . . . . . . . . . . . . . . . . . . 19 ∈ V
107106elixp 7866 . . . . . . . . . . . . . . . . . 18 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ↔ ( Fn 𝐴 ∧ ∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)))
108 ixpin 7884 . . . . . . . . . . . . . . . . . . . 20 X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) = (X𝑘𝐴 (𝑔𝑘) ∩ X𝑘𝐴 𝑆)
10961ineq1i 3793 . . . . . . . . . . . . . . . . . . . 20 (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) = (X𝑘𝐴 (𝑔𝑘) ∩ X𝑘𝐴 𝑆)
110108, 109eqtr4i 2646 . . . . . . . . . . . . . . . . . . 19 X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) = (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆)
111110neeq1i 2854 . . . . . . . . . . . . . . . . . 18 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅ ↔ (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
112105, 107, 1113imtr3i 280 . . . . . . . . . . . . . . . . 17 (( Fn 𝐴 ∧ ∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
113104, 112sylan2br 493 . . . . . . . . . . . . . . . 16 (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
11499, 113sylan 488 . . . . . . . . . . . . . . 15 ((:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
115114exlimiv 1855 . . . . . . . . . . . . . 14 (∃(:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
11698, 115syl 17 . . . . . . . . . . . . 13 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
117116expr 642 . . . . . . . . . . . 12 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)) → (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
11845, 117syldan 487 . . . . . . . . . . 11 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦))) → (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
1191183adantr3 1220 . . . . . . . . . 10 (((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 2687 . . . . . . . . . . 11 (𝑢 = X𝑦𝐴 (𝑔𝑦) → (𝑓𝑢𝑓X𝑦𝐴 (𝑔𝑦)))
121 ineq1 3790 . . . . . . . . . . . 12 (𝑢 = X𝑦𝐴 (𝑔𝑦) → (𝑢X𝑘𝐴 𝑆) = (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆))
122121neeq1d 2849 . . . . . . . . . . 11 (𝑢 = X𝑦𝐴 (𝑔𝑦) → ((𝑢X𝑘𝐴 𝑆) ≠ ∅ ↔ (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
123120, 122imbi12d 334 . . . . . . . . . 10 (𝑢 = X𝑦𝐴 (𝑔𝑦) → ((𝑓𝑢 → (𝑢X𝑘𝐴 𝑆) ≠ ∅) ↔ (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)))
124119, 123syl5ibrcom 237 . . . . . . . . 9 (((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 628 . . . . . . . 8 ((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 1858 . . . . . . 7 ((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 232 . . . . . 6 ((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 2960 . . . . 5 ((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) ≠ ∅))
1294, 37fmptd 6346 . . . . . . . . . 10 (𝜑 → (𝑘𝐴𝑅):𝐴⟶Top)
130 ffn 6007 . . . . . . . . . 10 ((𝑘𝐴𝑅):𝐴⟶Top → (𝑘𝐴𝑅) Fn 𝐴)
131129, 130syl 17 . . . . . . . . 9 (𝜑 → (𝑘𝐴𝑅) Fn 𝐴)
132 eqid 2621 . . . . . . . . . 10 {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))}
133132ptval 21292 . . . . . . . . 9 ((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))}))
1341, 131, 133syl2anc 692 . . . . . . . 8 (𝜑 → (∏t‘(𝑘𝐴𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
13513, 134syl5eq 2667 . . . . . . 7 (𝜑𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
136135adantr 481 . . . . . 6 ((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))}))
1372ralrimiva 2961 . . . . . . . . 9 (𝜑 → ∀𝑘𝐴 𝑅 ∈ (TopOn‘𝑋))
13813pttopon 21318 . . . . . . . . 9 ((𝐴𝑉 ∧ ∀𝑘𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋))
1391, 137, 138syl2anc 692 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋))
140 toponuni 20647 . . . . . . . 8 (𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋) → X𝑘𝐴 𝑋 = 𝐽)
141139, 140syl 17 . . . . . . 7 (𝜑X𝑘𝐴 𝑋 = 𝐽)
142141adantr 481 . . . . . 6 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘𝐴 𝑋 = 𝐽)
143132ptbas 21301 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑘𝐴𝑅):𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
1441, 129, 143syl2anc 692 . . . . . . 7 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
145144adantr 481 . . . . . 6 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
1465ralrimiva 2961 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 𝑆𝑋)
147 ss2ixp 7872 . . . . . . . 8 (∀𝑘𝐴 𝑆𝑋X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
148146, 147syl 17 . . . . . . 7 (𝜑X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
149148adantr 481 . . . . . 6 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
1509clsss3 20782 . . . . . . . . . . 11 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((cls‘𝑅)‘𝑆) ⊆ 𝑅)
1514, 8, 150syl2anc 692 . . . . . . . . . 10 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ⊆ 𝑅)
152151, 7sseqtr4d 3626 . . . . . . . . 9 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
153152ralrimiva 2961 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
154 ss2ixp 7872 . . . . . . . 8 (∀𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋X𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ X𝑘𝐴 𝑋)
155153, 154syl 17 . . . . . . 7 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ X𝑘𝐴 𝑋)
156155sselda 3587 . . . . . 6 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓X𝑘𝐴 𝑋)
157136, 142, 145, 149, 156elcls3 20806 . . . . 5 ((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) ≠ ∅)))
158128, 157mpbird 247 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓 ∈ ((cls‘𝐽)‘X𝑘𝐴 𝑆))
159158ex 450 . . 3 (𝜑 → (𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆) → 𝑓 ∈ ((cls‘𝐽)‘X𝑘𝐴 𝑆)))
160159ssrdv 3593 . 2 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ ((cls‘𝐽)‘X𝑘𝐴 𝑆))
16123, 160eqssd 3604 1 (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) = X𝑘𝐴 ((cls‘𝑅)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  {crab 2911  csb 3518  cdif 3556  cin 3558  wss 3559  c0 3896   cuni 4407   ciun 4490  cmpt 4678   Fn wfn 5847  wf 5848  cfv 5852  Xcixp 7859  Fincfn 7906  AC wacn 8715  topGenctg 16026  tcpt 16027  Topctop 20626  TopOnctopon 20643  TopBasesctb 20669  Clsdccld 20739  clsccl 20741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-ixp 7860  df-en 7907  df-fin 7910  df-fi 8268  df-acn 8719  df-topgen 16032  df-pt 16033  df-top 20627  df-topon 20644  df-bases 20670  df-cld 20742  df-ntr 20743  df-cls 20744
This theorem is referenced by:  ptcls  21338  dfac14  21340
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