MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac14 Structured version   Visualization version   GIF version

Theorem dfac14 21642
Description: Theorem ptcls 21640 is an equivalent of the axiom of choice. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
dfac14 (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))))
Distinct variable group:   𝑓,𝑘,𝑠

Proof of Theorem dfac14
Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6332 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝑓𝑘) = (𝑓𝑥))
21unieqd 4584 . . . . . . . . 9 (𝑘 = 𝑥 (𝑓𝑘) = (𝑓𝑥))
32pweqd 4302 . . . . . . . 8 (𝑘 = 𝑥 → 𝒫 (𝑓𝑘) = 𝒫 (𝑓𝑥))
43cbvixpv 8080 . . . . . . 7 X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) = X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)
54eleq2i 2842 . . . . . 6 (𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) ↔ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥))
6 simplr 752 . . . . . . . . . . 11 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑓:dom 𝑓⟶Top)
76feqmptd 6391 . . . . . . . . . 10 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑓 = (𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘)))
87fveq2d 6336 . . . . . . . . 9 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → (∏t𝑓) = (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘))))
98fveq2d 6336 . . . . . . . 8 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → (cls‘(∏t𝑓)) = (cls‘(∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘)))))
109fveq1d 6334 . . . . . . 7 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → ((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = ((cls‘(∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘))))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)))
11 eqid 2771 . . . . . . . 8 (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘))) = (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘)))
12 vex 3354 . . . . . . . . . 10 𝑓 ∈ V
1312dmex 7246 . . . . . . . . 9 dom 𝑓 ∈ V
1413a1i 11 . . . . . . . 8 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → dom 𝑓 ∈ V)
156ffvelrnda 6502 . . . . . . . . 9 ((((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑓𝑘) ∈ Top)
16 eqid 2771 . . . . . . . . . 10 (𝑓𝑘) = (𝑓𝑘)
1716toptopon 20942 . . . . . . . . 9 ((𝑓𝑘) ∈ Top ↔ (𝑓𝑘) ∈ (TopOn‘ (𝑓𝑘)))
1815, 17sylib 208 . . . . . . . 8 ((((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑓𝑘) ∈ (TopOn‘ (𝑓𝑘)))
19 simpr 471 . . . . . . . . . . . 12 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥))
2019, 5sylibr 224 . . . . . . . . . . 11 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘))
21 vex 3354 . . . . . . . . . . . . 13 𝑠 ∈ V
2221elixp 8069 . . . . . . . . . . . 12 (𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) ↔ (𝑠 Fn dom 𝑓 ∧ ∀𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ 𝒫 (𝑓𝑘)))
2322simprbi 484 . . . . . . . . . . 11 (𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) → ∀𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ 𝒫 (𝑓𝑘))
2420, 23syl 17 . . . . . . . . . 10 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → ∀𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ 𝒫 (𝑓𝑘))
2524r19.21bi 3081 . . . . . . . . 9 ((((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑠𝑘) ∈ 𝒫 (𝑓𝑘))
2625elpwid 4309 . . . . . . . 8 ((((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑠𝑘) ⊆ (𝑓𝑘))
27 fvex 6342 . . . . . . . . . 10 (𝑠𝑘) ∈ V
2813, 27iunex 7294 . . . . . . . . 9 𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ V
29 simpll 750 . . . . . . . . . 10 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → CHOICE)
30 acacni 9164 . . . . . . . . . 10 ((CHOICE ∧ dom 𝑓 ∈ V) → AC dom 𝑓 = V)
3129, 13, 30sylancl 574 . . . . . . . . 9 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → AC dom 𝑓 = V)
3228, 31syl5eleqr 2857 . . . . . . . 8 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ AC dom 𝑓)
3311, 14, 18, 26, 32ptclsg 21639 . . . . . . 7 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → ((cls‘(∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘))))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)))
3410, 33eqtrd 2805 . . . . . 6 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → ((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)))
355, 34sylan2b 581 . . . . 5 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)) → ((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)))
3635ralrimiva 3115 . . . 4 ((CHOICE𝑓:dom 𝑓⟶Top) → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)))
3736ex 397 . . 3 (CHOICE → (𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))))
3837alrimiv 2007 . 2 (CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))))
39 vex 3354 . . . . . . . 8 𝑔 ∈ V
4039dmex 7246 . . . . . . 7 dom 𝑔 ∈ V
4140a1i 11 . . . . . 6 ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → dom 𝑔 ∈ V)
42 fvex 6342 . . . . . . 7 (𝑔𝑥) ∈ V
4342a1i 11 . . . . . 6 (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ V)
44 simplrr 763 . . . . . . . 8 (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → ∅ ∉ ran 𝑔)
45 df-nel 3047 . . . . . . . 8 (∅ ∉ ran 𝑔 ↔ ¬ ∅ ∈ ran 𝑔)
4644, 45sylib 208 . . . . . . 7 (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔)
47 funforn 6263 . . . . . . . . . . . 12 (Fun 𝑔𝑔:dom 𝑔onto→ran 𝑔)
48 fof 6256 . . . . . . . . . . . 12 (𝑔:dom 𝑔onto→ran 𝑔𝑔:dom 𝑔⟶ran 𝑔)
4947, 48sylbi 207 . . . . . . . . . . 11 (Fun 𝑔𝑔:dom 𝑔⟶ran 𝑔)
5049ad2antrl 707 . . . . . . . . . 10 ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → 𝑔:dom 𝑔⟶ran 𝑔)
5150ffvelrnda 6502 . . . . . . . . 9 (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ ran 𝑔)
52 eleq1 2838 . . . . . . . . 9 ((𝑔𝑥) = ∅ → ((𝑔𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔))
5351, 52syl5ibcom 235 . . . . . . . 8 (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔𝑥) = ∅ → ∅ ∈ ran 𝑔))
5453necon3bd 2957 . . . . . . 7 (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔𝑥) ≠ ∅))
5546, 54mpd 15 . . . . . 6 (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ≠ ∅)
56 eqid 2771 . . . . . 6 𝒫 (𝑔𝑥) = 𝒫 (𝑔𝑥)
57 eqid 2771 . . . . . 6 {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} = {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}
58 eqid 2771 . . . . . 6 (∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}))
59 fveq1 6331 . . . . . . . . . . 11 (𝑠 = 𝑔 → (𝑠𝑘) = (𝑔𝑘))
6059ixpeq2dv 8078 . . . . . . . . . 10 (𝑠 = 𝑔X𝑘 ∈ dom 𝑔(𝑠𝑘) = X𝑘 ∈ dom 𝑔(𝑔𝑘))
61 fveq2 6332 . . . . . . . . . . 11 (𝑘 = 𝑥 → (𝑔𝑘) = (𝑔𝑥))
6261cbvixpv 8080 . . . . . . . . . 10 X𝑘 ∈ dom 𝑔(𝑔𝑘) = X𝑥 ∈ dom 𝑔(𝑔𝑥)
6360, 62syl6eq 2821 . . . . . . . . 9 (𝑠 = 𝑔X𝑘 ∈ dom 𝑔(𝑠𝑘) = X𝑥 ∈ dom 𝑔(𝑔𝑥))
6463fveq2d 6336 . . . . . . . 8 (𝑠 = 𝑔 → ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠𝑘)) = ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑥 ∈ dom 𝑔(𝑔𝑥)))
6559fveq2d 6336 . . . . . . . . . 10 (𝑠 = 𝑔 → ((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘)) = ((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑔𝑘)))
6665ixpeq2dv 8078 . . . . . . . . 9 (𝑠 = 𝑔X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑔𝑘)))
6761unieqd 4584 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 (𝑔𝑘) = (𝑔𝑥))
6867pweqd 4302 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑥 → 𝒫 (𝑔𝑘) = 𝒫 (𝑔𝑥))
6968sneqd 4328 . . . . . . . . . . . . . . 15 (𝑘 = 𝑥 → {𝒫 (𝑔𝑘)} = {𝒫 (𝑔𝑥)})
7061, 69uneq12d 3919 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))
7170pweqd 4302 . . . . . . . . . . . . 13 (𝑘 = 𝑥 → 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) = 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))
7268eleq1d 2835 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → (𝒫 (𝑔𝑘) ∈ 𝑦 ↔ 𝒫 (𝑔𝑥) ∈ 𝑦))
7370eqeq2d 2781 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → (𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ↔ 𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})))
7472, 73imbi12d 333 . . . . . . . . . . . . 13 (𝑘 = 𝑥 → ((𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})) ↔ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))))
7571, 74rabeqbidv 3345 . . . . . . . . . . . 12 (𝑘 = 𝑥 → {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))} = {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})
7675fveq2d 6336 . . . . . . . . . . 11 (𝑘 = 𝑥 → (cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))}) = (cls‘{𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}))
7776, 61fveq12d 6338 . . . . . . . . . 10 (𝑘 = 𝑥 → ((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑔𝑘)) = ((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})‘(𝑔𝑥)))
7877cbvixpv 8080 . . . . . . . . 9 X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑔𝑘)) = X𝑥 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})‘(𝑔𝑥))
7966, 78syl6eq 2821 . . . . . . . 8 (𝑠 = 𝑔X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘)) = X𝑥 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})‘(𝑔𝑥)))
8064, 79eqeq12d 2786 . . . . . . 7 (𝑠 = 𝑔 → (((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘)) ↔ ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑥 ∈ dom 𝑔(𝑔𝑥)) = X𝑥 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})‘(𝑔𝑥))))
81 simpl 468 . . . . . . . 8 ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → ∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))))
82 snex 5036 . . . . . . . . . . . . 13 {𝒫 (𝑔𝑥)} ∈ V
8342, 82unex 7103 . . . . . . . . . . . 12 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∈ V
84 ssun2 3928 . . . . . . . . . . . . 13 {𝒫 (𝑔𝑥)} ⊆ ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})
8542uniex 7100 . . . . . . . . . . . . . . 15 (𝑔𝑥) ∈ V
8685pwex 4981 . . . . . . . . . . . . . 14 𝒫 (𝑔𝑥) ∈ V
8786snid 4347 . . . . . . . . . . . . 13 𝒫 (𝑔𝑥) ∈ {𝒫 (𝑔𝑥)}
8884, 87sselii 3749 . . . . . . . . . . . 12 𝒫 (𝑔𝑥) ∈ ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})
89 epttop 21034 . . . . . . . . . . . 12 ((((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∈ V ∧ 𝒫 (𝑔𝑥) ∈ ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})) → {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} ∈ (TopOn‘((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})))
9083, 88, 89mp2an 672 . . . . . . . . . . 11 {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} ∈ (TopOn‘((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))
9190topontopi 20940 . . . . . . . . . 10 {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} ∈ Top
9291a1i 11 . . . . . . . . 9 (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} ∈ Top)
93 eqid 2771 . . . . . . . . 9 (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})
9492, 93fmptd 6527 . . . . . . . 8 ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}):dom 𝑔⟶Top)
9540mptex 6630 . . . . . . . . 9 (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) ∈ V
96 id 22 . . . . . . . . . . 11 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → 𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}))
97 dmeq 5462 . . . . . . . . . . . 12 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → dom 𝑓 = dom (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}))
9883pwex 4981 . . . . . . . . . . . . . 14 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∈ V
9998rabex 4946 . . . . . . . . . . . . 13 {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} ∈ V
10099, 93dmmpti 6163 . . . . . . . . . . . 12 dom (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) = dom 𝑔
10197, 100syl6eq 2821 . . . . . . . . . . 11 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → dom 𝑓 = dom 𝑔)
10296, 101feq12d 6173 . . . . . . . . . 10 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → (𝑓:dom 𝑓⟶Top ↔ (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}):dom 𝑔⟶Top))
103101ixpeq1d 8074 . . . . . . . . . . . 12 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) = X𝑘 ∈ dom 𝑔𝒫 (𝑓𝑘))
104 fveq1 6331 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → (𝑓𝑘) = ((𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})‘𝑘))
105 fveq2 6332 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑘 → (𝑔𝑥) = (𝑔𝑘))
106105unieqd 4584 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘 (𝑔𝑥) = (𝑔𝑘))
107106pweqd 4302 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑘 → 𝒫 (𝑔𝑥) = 𝒫 (𝑔𝑘))
108107sneqd 4328 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑘 → {𝒫 (𝑔𝑥)} = {𝒫 (𝑔𝑘)})
109105, 108uneq12d 3919 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
110109pweqd 4302 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑘 → 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) = 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
111107eleq1d 2835 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → (𝒫 (𝑔𝑥) ∈ 𝑦 ↔ 𝒫 (𝑔𝑘) ∈ 𝑦))
112109eqeq2d 2781 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → (𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ↔ 𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})))
113111, 112imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑘 → ((𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})) ↔ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))))
114110, 113rabeqbidv 3345 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑘 → {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} = {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})
115 fvex 6342 . . . . . . . . . . . . . . . . . . . . 21 (𝑔𝑘) ∈ V
116 snex 5036 . . . . . . . . . . . . . . . . . . . . 21 {𝒫 (𝑔𝑘)} ∈ V
117115, 116unex 7103 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∈ V
118117pwex 4981 . . . . . . . . . . . . . . . . . . 19 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∈ V
119118rabex 4946 . . . . . . . . . . . . . . . . . 18 {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))} ∈ V
120114, 93, 119fvmpt 6424 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ dom 𝑔 → ((𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})‘𝑘) = {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})
121104, 120sylan9eq 2825 . . . . . . . . . . . . . . . 16 ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → (𝑓𝑘) = {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})
122121unieqd 4584 . . . . . . . . . . . . . . 15 ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → (𝑓𝑘) = {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})
123 ssun2 3928 . . . . . . . . . . . . . . . . . 18 {𝒫 (𝑔𝑘)} ⊆ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
124115uniex 7100 . . . . . . . . . . . . . . . . . . . 20 (𝑔𝑘) ∈ V
125124pwex 4981 . . . . . . . . . . . . . . . . . . 19 𝒫 (𝑔𝑘) ∈ V
126125snid 4347 . . . . . . . . . . . . . . . . . 18 𝒫 (𝑔𝑘) ∈ {𝒫 (𝑔𝑘)}
127123, 126sselii 3749 . . . . . . . . . . . . . . . . 17 𝒫 (𝑔𝑘) ∈ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
128 epttop 21034 . . . . . . . . . . . . . . . . 17 ((((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∈ V ∧ 𝒫 (𝑔𝑘) ∈ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})) → {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))} ∈ (TopOn‘((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})))
129117, 127, 128mp2an 672 . . . . . . . . . . . . . . . 16 {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))} ∈ (TopOn‘((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
130129toponunii 20941 . . . . . . . . . . . . . . 15 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) = {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))}
131122, 130syl6eqr 2823 . . . . . . . . . . . . . 14 ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → (𝑓𝑘) = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
132131pweqd 4302 . . . . . . . . . . . . 13 ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → 𝒫 (𝑓𝑘) = 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
133132ixpeq2dva 8077 . . . . . . . . . . . 12 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → X𝑘 ∈ dom 𝑔𝒫 (𝑓𝑘) = X𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
134103, 133eqtrd 2805 . . . . . . . . . . 11 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) = X𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
135 fveq2 6332 . . . . . . . . . . . . . 14 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → (∏t𝑓) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))
136135fveq2d 6336 . . . . . . . . . . . . 13 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → (cls‘(∏t𝑓)) = (cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}))))
137101ixpeq1d 8074 . . . . . . . . . . . . 13 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → X𝑘 ∈ dom 𝑓(𝑠𝑘) = X𝑘 ∈ dom 𝑔(𝑠𝑘))
138136, 137fveq12d 6338 . . . . . . . . . . . 12 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → ((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠𝑘)))
139101ixpeq1d 8074 . . . . . . . . . . . . 13 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)) = X𝑘 ∈ dom 𝑔((cls‘(𝑓𝑘))‘(𝑠𝑘)))
140121fveq2d 6336 . . . . . . . . . . . . . . 15 ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → (cls‘(𝑓𝑘)) = (cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))}))
141140fveq1d 6334 . . . . . . . . . . . . . 14 ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → ((cls‘(𝑓𝑘))‘(𝑠𝑘)) = ((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘)))
142141ixpeq2dva 8077 . . . . . . . . . . . . 13 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → X𝑘 ∈ dom 𝑔((cls‘(𝑓𝑘))‘(𝑠𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘)))
143139, 142eqtrd 2805 . . . . . . . . . . . 12 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘)))
144138, 143eqeq12d 2786 . . . . . . . . . . 11 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → (((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)) ↔ ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘))))
145134, 144raleqbidv 3301 . . . . . . . . . 10 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → (∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)) ↔ ∀𝑠X 𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘))))
146102, 145imbi12d 333 . . . . . . . . 9 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → ((𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ↔ ((𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}):dom 𝑔⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘)))))
14795, 146spcv 3450 . . . . . . . 8 (∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) → ((𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}):dom 𝑔⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘))))
14881, 94, 147sylc 65 . . . . . . 7 ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → ∀𝑠X 𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))})‘(𝑠𝑘)))
149 simprl 754 . . . . . . . . 9 ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → Fun 𝑔)
150 funfn 6061 . . . . . . . . 9 (Fun 𝑔𝑔 Fn dom 𝑔)
151149, 150sylib 208 . . . . . . . 8 ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → 𝑔 Fn dom 𝑔)
152 ssun1 3927 . . . . . . . . . 10 (𝑔𝑘) ⊆ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
153115elpw 4303 . . . . . . . . . 10 ((𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ↔ (𝑔𝑘) ⊆ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
154152, 153mpbir 221 . . . . . . . . 9 (𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
155154rgenw 3073 . . . . . . . 8 𝑘 ∈ dom 𝑔(𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
15639elixp 8069 . . . . . . . 8 (𝑔X𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ↔ (𝑔 Fn dom 𝑔 ∧ ∀𝑘 ∈ dom 𝑔(𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})))
157151, 155, 156sylanblrc 578 . . . . . . 7 ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → 𝑔X𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
15880, 148, 157rspcdva 3466 . . . . . 6 ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})))‘X𝑥 ∈ dom 𝑔(𝑔𝑥)) = X𝑥 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})‘(𝑔𝑥)))
15941, 43, 55, 56, 57, 58, 158dfac14lem 21641 . . . . 5 ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅)
160159ex 397 . . . 4 (∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) → ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
161160alrimiv 2007 . . 3 (∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
162 dfac9 9160 . . 3 (CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
163161, 162sylibr 224 . 2 (∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))) → CHOICE)
16438, 163impbii 199 1 (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wcel 2145  wne 2943  wnel 3046  wral 3061  {crab 3065  Vcvv 3351  cun 3721  wss 3723  c0 4063  𝒫 cpw 4297  {csn 4316   cuni 4574   ciun 4654  cmpt 4863  dom cdm 5249  ran crn 5250  Fun wfun 6025   Fn wfn 6026  wf 6027  ontowfo 6029  cfv 6031  Xcixp 8062  AC wacn 8964  CHOICEwac 9138  tcpt 16307  Topctop 20918  TopOnctopon 20935  clsccl 21043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-ixp 8063  df-en 8110  df-dom 8111  df-fin 8113  df-fi 8473  df-card 8965  df-acn 8968  df-ac 9139  df-topgen 16312  df-pt 16313  df-top 20919  df-topon 20936  df-bases 20971  df-cld 21044  df-ntr 21045  df-cls 21046
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator