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Theorem ptclsg 23623
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 22919 . . . . . 6 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
42, 3syl 17 . . . . 5 ((𝜑𝑘𝐴) → 𝑅 ∈ Top)
5 ptcls.c . . . . . . 7 ((𝜑𝑘𝐴) → 𝑆𝑋)
6 toponuni 22920 . . . . . . . 8 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
72, 6syl 17 . . . . . . 7 ((𝜑𝑘𝐴) → 𝑋 = 𝑅)
85, 7sseqtrd 4020 . . . . . 6 ((𝜑𝑘𝐴) → 𝑆 𝑅)
9 eqid 2737 . . . . . . 7 𝑅 = 𝑅
109clscld 23055 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
114, 8, 10syl2anc 584 . . . . 5 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝑅))
121, 4, 11ptcldmpt 23622 . . . 4 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘(∏t‘(𝑘𝐴𝑅))))
13 ptcls.2 . . . . 5 𝐽 = (∏t‘(𝑘𝐴𝑅))
1413fveq2i 6909 . . . 4 (Clsd‘𝐽) = (Clsd‘(∏t‘(𝑘𝐴𝑅)))
1512, 14eleqtrrdi 2852 . . 3 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽))
169sscls 23064 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
174, 8, 16syl2anc 584 . . . . 5 ((𝜑𝑘𝐴) → 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
1817ralrimiva 3146 . . . 4 (𝜑 → ∀𝑘𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆))
19 ss2ixp 8950 . . . 4 (∀𝑘𝐴 𝑆 ⊆ ((cls‘𝑅)‘𝑆) → X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆))
2018, 19syl 17 . . 3 (𝜑X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆))
21 eqid 2737 . . . 4 𝐽 = 𝐽
2221clsss2 23080 . . 3 ((X𝑘𝐴 ((cls‘𝑅)‘𝑆) ∈ (Clsd‘𝐽) ∧ X𝑘𝐴 𝑆X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → ((cls‘𝐽)‘X𝑘𝐴 𝑆) ⊆ X𝑘𝐴 ((cls‘𝑅)‘𝑆))
2315, 20, 22syl2anc 584 . 2 (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) ⊆ X𝑘𝐴 ((cls‘𝑅)‘𝑆))
24 vex 3484 . . . . . 6 𝑢 ∈ V
25 eqeq1 2741 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑢 = X𝑦𝐴 (𝑔𝑦)))
2625anbi2d 630 . . . . . . 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 1921 . . . . . 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 3679 . . . . 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 6917 . . . . . . . . . . . . . . . 16 𝑘((𝑘𝐴𝑅)‘𝑦)
3029nfel2 2924 . . . . . . . . . . . . . . 15 𝑘(𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)
31 nfv 1914 . . . . . . . . . . . . . . 15 𝑦(𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘)
32 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑔𝑦) = (𝑔𝑘))
33 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → ((𝑘𝐴𝑅)‘𝑦) = ((𝑘𝐴𝑅)‘𝑘))
3432, 33eleq12d 2835 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → ((𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘)))
3530, 31, 34cbvralw 3306 . . . . . . . . . . . . . 14 (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘))
36 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → 𝑘𝐴)
37 eqid 2737 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴𝑅) = (𝑘𝐴𝑅)
3837fvmpt2 7027 . . . . . . . . . . . . . . . . 17 ((𝑘𝐴𝑅 ∈ (TopOn‘𝑋)) → ((𝑘𝐴𝑅)‘𝑘) = 𝑅)
3936, 2, 38syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑅)‘𝑘) = 𝑅)
4039eleq2d 2827 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → ((𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘) ↔ (𝑔𝑘) ∈ 𝑅))
4140ralbidva 3176 . . . . . . . . . . . . . 14 (𝜑 → (∀𝑘𝐴 (𝑔𝑘) ∈ ((𝑘𝐴𝑅)‘𝑘) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅))
4235, 41bitrid 283 . . . . . . . . . . . . 13 (𝜑 → (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅))
4342anbi2d 630 . . . . . . . . . . . 12 (𝜑 → ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)))
4443adantr 480 . . . . . . . . . . 11 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)))
4544biimpa 476 . . . . . . . . . 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 726 . . . . . . . . . . . . 13 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑘𝐴 𝑆AC 𝐴)
48 simpll 767 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝜑)
49 vex 3484 . . . . . . . . . . . . . . . . . . . 20 𝑓 ∈ V
5049elixp 8944 . . . . . . . . . . . . . . . . . . 19 (𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)))
5150simprbi 496 . . . . . . . . . . . . . . . . . 18 (𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆) → ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆))
5251ad2antlr 727 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆))
539clsndisj 23083 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 𝑅 ∧ (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)) ∧ ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘))) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
5453ex 412 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Top ∧ 𝑆 𝑅 ∧ (𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆)) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
55543expia 1122 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)))
564, 8, 55syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → ((𝑓𝑘) ∈ ((cls‘𝑅)‘𝑆) → (((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) → ((𝑔𝑘) ∩ 𝑆) ≠ ∅)))
5756ralimdva 3167 . . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)
60 simprr 773 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑓X𝑦𝐴 (𝑔𝑦))
6132cbvixpv 8955 . . . . . . . . . . . . . . . . . . 19 X𝑦𝐴 (𝑔𝑦) = X𝑘𝐴 (𝑔𝑘)
6260, 61eleqtrdi 2851 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → 𝑓X𝑘𝐴 (𝑔𝑘))
6349elixp 8944 . . . . . . . . . . . . . . . . . . 19 (𝑓X𝑘𝐴 (𝑔𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘)))
6463simprbi 496 . . . . . . . . . . . . . . . . . 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 3111 . . . . . . . . . . . . . . . . 17 (∀𝑘𝐴 ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)) ↔ (∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝑔𝑘)))
6759, 65, 66sylanbrc 583 . . . . . . . . . . . . . . . 16 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴 ((𝑔𝑘) ∈ 𝑅 ∧ (𝑓𝑘) ∈ (𝑔𝑘)))
68 ralim 3086 . . . . . . . . . . . . . . . 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 4389 . . . . . . . . . . . . . . . . 17 ({𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} ≠ ∅ ↔ ∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆))
71 dfin5 3959 . . . . . . . . . . . . . . . . . . 19 ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = {𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)}
72 inss2 4238 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔𝑘) ∩ 𝑆) ⊆ 𝑆
73 ssiun2 5047 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝐴𝑆 𝑘𝐴 𝑆)
7472, 73sstrid 3995 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐴 → ((𝑔𝑘) ∩ 𝑆) ⊆ 𝑘𝐴 𝑆)
75 sseqin2 4223 . . . . . . . . . . . . . . . . . . . 20 (((𝑔𝑘) ∩ 𝑆) ⊆ 𝑘𝐴 𝑆 ↔ ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = ((𝑔𝑘) ∩ 𝑆))
7674, 75sylib 218 . . . . . . . . . . . . . . . . . . 19 (𝑘𝐴 → ( 𝑘𝐴 𝑆 ∩ ((𝑔𝑘) ∩ 𝑆)) = ((𝑔𝑘) ∩ 𝑆))
7771, 76eqtr3id 2791 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴 → {𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} = ((𝑔𝑘) ∩ 𝑆))
7877neeq1d 3000 . . . . . . . . . . . . . . . . 17 (𝑘𝐴 → ({𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)} ≠ ∅ ↔ ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
7970, 78bitr3id 285 . . . . . . . . . . . . . . . 16 (𝑘𝐴 → (∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ((𝑔𝑘) ∩ 𝑆) ≠ ∅))
8079ralbiia 3091 . . . . . . . . . . . . . . 15 (∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
8169, 80sylibr 234 . . . . . . . . . . . . . 14 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆))
82 nfv 1914 . . . . . . . . . . . . . . 15 𝑦𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆)
83 nfiu1 5027 . . . . . . . . . . . . . . . 16 𝑘 𝑘𝐴 𝑆
84 nfcv 2905 . . . . . . . . . . . . . . . . . 18 𝑘(𝑔𝑦)
85 nfcsb1v 3923 . . . . . . . . . . . . . . . . . 18 𝑘𝑦 / 𝑘𝑆
8684, 85nfin 4224 . . . . . . . . . . . . . . . . 17 𝑘((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
8786nfel2 2924 . . . . . . . . . . . . . . . 16 𝑘 𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
8883, 87nfrexw 3313 . . . . . . . . . . . . . . 15 𝑘𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
89 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦 → (𝑔𝑘) = (𝑔𝑦))
90 csbeq1a 3913 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦𝑆 = 𝑦 / 𝑘𝑆)
9189, 90ineq12d 4221 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → ((𝑔𝑘) ∩ 𝑆) = ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
9291eleq2d 2827 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → (𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ 𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9392rexbidv 3179 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∃𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9482, 88, 93cbvralw 3306 . . . . . . . . . . . . . 14 (∀𝑘𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
9581, 94sylib 218 . . . . . . . . . . . . 13 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
96 eleq1 2829 . . . . . . . . . . . . . 14 (𝑧 = (𝑦) → (𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆) ↔ (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9796acni3 10087 . . . . . . . . . . . . 13 (( 𝑘𝐴 𝑆AC 𝐴 ∧ ∀𝑦𝐴𝑧 𝑘𝐴 𝑆𝑧 ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → ∃(:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
9847, 95, 97syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ ((𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅) ∧ 𝑓X𝑦𝐴 (𝑔𝑦))) → ∃(:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
99 ffn 6736 . . . . . . . . . . . . . 14 (:𝐴 𝑘𝐴 𝑆 Fn 𝐴)
100 nfv 1914 . . . . . . . . . . . . . . . 16 𝑦(𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)
10186nfel2 2924 . . . . . . . . . . . . . . . 16 𝑘(𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)
102 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → (𝑘) = (𝑦))
103102, 91eleq12d 2835 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → ((𝑘) ∈ ((𝑔𝑘) ∩ 𝑆) ↔ (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)))
104100, 101, 103cbvralw 3306 . . . . . . . . . . . . . . 15 (∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆) ↔ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆))
105 ne0i 4341 . . . . . . . . . . . . . . . 16 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) → X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅)
106 vex 3484 . . . . . . . . . . . . . . . . 17 ∈ V
107106elixp 8944 . . . . . . . . . . . . . . . 16 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ↔ ( Fn 𝐴 ∧ ∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)))
108 ixpin 8963 . . . . . . . . . . . . . . . . . 18 X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) = (X𝑘𝐴 (𝑔𝑘) ∩ X𝑘𝐴 𝑆)
10961ineq1i 4216 . . . . . . . . . . . . . . . . . 18 (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) = (X𝑘𝐴 (𝑔𝑘) ∩ X𝑘𝐴 𝑆)
110108, 109eqtr4i 2768 . . . . . . . . . . . . . . . . 17 X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) = (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆)
111110neeq1i 3005 . . . . . . . . . . . . . . . 16 (X𝑘𝐴 ((𝑔𝑘) ∩ 𝑆) ≠ ∅ ↔ (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
112105, 107, 1113imtr3i 291 . . . . . . . . . . . . . . 15 (( Fn 𝐴 ∧ ∀𝑘𝐴 (𝑘) ∈ ((𝑔𝑘) ∩ 𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
113104, 112sylan2br 595 . . . . . . . . . . . . . 14 (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
11499, 113sylan 580 . . . . . . . . . . . . 13 ((:𝐴 𝑘𝐴 𝑆 ∧ ∀𝑦𝐴 (𝑦) ∈ ((𝑔𝑦) ∩ 𝑦 / 𝑘𝑆)) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)
115114exlimiv 1930 . . . . . . . . . . . 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 456 . . . . . . . . . 10 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑅)) → (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
11845, 117syldan 591 . . . . . . . . 9 (((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦))) → (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
1191183adantr3 1172 . . . . . . . 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 2830 . . . . . . . . 9 (𝑢 = X𝑦𝐴 (𝑔𝑦) → (𝑓𝑢𝑓X𝑦𝐴 (𝑔𝑦)))
121 ineq1 4213 . . . . . . . . . 10 (𝑢 = X𝑦𝐴 (𝑔𝑦) → (𝑢X𝑘𝐴 𝑆) = (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆))
122121neeq1d 3000 . . . . . . . . 9 (𝑢 = X𝑦𝐴 (𝑔𝑦) → ((𝑢X𝑘𝐴 𝑆) ≠ ∅ ↔ (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅))
123120, 122imbi12d 344 . . . . . . . 8 (𝑢 = X𝑦𝐴 (𝑔𝑦) → ((𝑓𝑢 → (𝑢X𝑘𝐴 𝑆) ≠ ∅) ↔ (𝑓X𝑦𝐴 (𝑔𝑦) → (X𝑦𝐴 (𝑔𝑦) ∩ X𝑘𝐴 𝑆) ≠ ∅)))
124119, 123syl5ibrcom 247 . . . . . . 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 453 . . . . . 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 1933 . . . . 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, 126biimtrid 242 . . . 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 3145 . . 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 7135 . . . . . . . 8 (𝜑 → (𝑘𝐴𝑅):𝐴⟶Top)
130129ffnd 6737 . . . . . . 7 (𝜑 → (𝑘𝐴𝑅) Fn 𝐴)
131 eqid 2737 . . . . . . . 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 23578 . . . . . . 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 584 . . . . . 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, 133eqtrid 2789 . . . . 5 (𝜑𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
135134adantr 480 . . . 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 3146 . . . . . . 7 (𝜑 → ∀𝑘𝐴 𝑅 ∈ (TopOn‘𝑋))
13713pttopon 23604 . . . . . . 7 ((𝐴𝑉 ∧ ∀𝑘𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋))
1381, 136, 137syl2anc 584 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋))
139 toponuni 22920 . . . . . 6 (𝐽 ∈ (TopOn‘X𝑘𝐴 𝑋) → X𝑘𝐴 𝑋 = 𝐽)
140138, 139syl 17 . . . . 5 (𝜑X𝑘𝐴 𝑋 = 𝐽)
141140adantr 480 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘𝐴 𝑋 = 𝐽)
142131ptbas 23587 . . . . . 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 584 . . . . 5 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝑘𝐴𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝑘𝐴𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
144143adantr 480 . . . 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 3146 . . . . . 6 (𝜑 → ∀𝑘𝐴 𝑆𝑋)
146 ss2ixp 8950 . . . . . 6 (∀𝑘𝐴 𝑆𝑋X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
147145, 146syl 17 . . . . 5 (𝜑X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
148147adantr 480 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → X𝑘𝐴 𝑆X𝑘𝐴 𝑋)
1499clsss3 23067 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 𝑅) → ((cls‘𝑅)‘𝑆) ⊆ 𝑅)
1504, 8, 149syl2anc 584 . . . . . . . 8 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ⊆ 𝑅)
151150, 7sseqtrrd 4021 . . . . . . 7 ((𝜑𝑘𝐴) → ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
152151ralrimiva 3146 . . . . . 6 (𝜑 → ∀𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋)
153 ss2ixp 8950 . . . . . 6 (∀𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ 𝑋X𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ X𝑘𝐴 𝑋)
154152, 153syl 17 . . . . 5 (𝜑X𝑘𝐴 ((cls‘𝑅)‘𝑆) ⊆ X𝑘𝐴 𝑋)
155154sselda 3983 . . . 4 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓X𝑘𝐴 𝑋)
156135, 141, 144, 148, 155elcls3 23091 . . 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 257 . 2 ((𝜑𝑓X𝑘𝐴 ((cls‘𝑅)‘𝑆)) → 𝑓 ∈ ((cls‘𝐽)‘X𝑘𝐴 𝑆))
15823, 157eqelssd 4005 1 (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) = X𝑘𝐴 ((cls‘𝑅)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  {crab 3436  csb 3899  cdif 3948  cin 3950  wss 3951  c0 4333   cuni 4907   ciun 4991  cmpt 5225   Fn wfn 6556  wf 6557  cfv 6561  Xcixp 8937  Fincfn 8985  AC wacn 9978  topGenctg 17482  tcpt 17483  Topctop 22899  TopOnctopon 22916  TopBasesctb 22952  Clsdccld 23024  clsccl 23026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1o 8506  df-2o 8507  df-map 8868  df-ixp 8938  df-en 8986  df-fin 8989  df-fi 9451  df-acn 9982  df-topgen 17488  df-pt 17489  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029
This theorem is referenced by:  ptcls  23624  dfac14  23626
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