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

Theorem dfac14 21469
Description: Theorem ptcls 21467 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 6229 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝑓𝑘) = (𝑓𝑥))
21unieqd 4478 . . . . . . . . 9 (𝑘 = 𝑥 (𝑓𝑘) = (𝑓𝑥))
32pweqd 4196 . . . . . . . 8 (𝑘 = 𝑥 → 𝒫 (𝑓𝑘) = 𝒫 (𝑓𝑥))
43cbvixpv 7968 . . . . . . 7 X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) = X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)
54eleq2i 2722 . . . . . 6 (𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) ↔ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥))
6 simplr 807 . . . . . . . . . . 11 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑓:dom 𝑓⟶Top)
76feqmptd 6288 . . . . . . . . . 10 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑓 = (𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘)))
87fveq2d 6233 . . . . . . . . 9 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → (∏t𝑓) = (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘))))
98fveq2d 6233 . . . . . . . 8 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → (cls‘(∏t𝑓)) = (cls‘(∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘)))))
109fveq1d 6231 . . . . . . 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 2651 . . . . . . . 8 (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘))) = (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘)))
12 vex 3234 . . . . . . . . . 10 𝑓 ∈ V
1312dmex 7141 . . . . . . . . 9 dom 𝑓 ∈ V
1413a1i 11 . . . . . . . 8 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → dom 𝑓 ∈ V)
156ffvelrnda 6399 . . . . . . . . 9 ((((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑓𝑘) ∈ Top)
16 eqid 2651 . . . . . . . . . 10 (𝑓𝑘) = (𝑓𝑘)
1716toptopon 20770 . . . . . . . . 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 476 . . . . . . . . . . . 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 3234 . . . . . . . . . . . . 13 𝑠 ∈ V
2221elixp 7957 . . . . . . . . . . . 12 (𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) ↔ (𝑠 Fn dom 𝑓 ∧ ∀𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ 𝒫 (𝑓𝑘)))
2322simprbi 479 . . . . . . . . . . 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 2961 . . . . . . . . 9 ((((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑠𝑘) ∈ 𝒫 (𝑓𝑘))
2625elpwid 4203 . . . . . . . 8 ((((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑠𝑘) ⊆ (𝑓𝑘))
27 fvex 6239 . . . . . . . . . 10 (𝑠𝑘) ∈ V
2813, 27iunex 7189 . . . . . . . . 9 𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ V
29 simpll 805 . . . . . . . . . 10 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → CHOICE)
30 acacni 9000 . . . . . . . . . 10 ((CHOICE ∧ dom 𝑓 ∈ V) → AC dom 𝑓 = V)
3129, 13, 30sylancl 695 . . . . . . . . 9 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → AC dom 𝑓 = V)
3228, 31syl5eleqr 2737 . . . . . . . 8 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ AC dom 𝑓)
3311, 14, 18, 26, 32ptclsg 21466 . . . . . . 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 2685 . . . . . 6 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → ((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)))
355, 34sylan2b 491 . . . . 5 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)) → ((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)))
3635ralrimiva 2995 . . . 4 ((CHOICE𝑓:dom 𝑓⟶Top) → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)))
3736ex 449 . . 3 (CHOICE → (𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))))
3837alrimiv 1895 . 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 3234 . . . . . . . 8 𝑔 ∈ V
4039dmex 7141 . . . . . . 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 6239 . . . . . . 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 818 . . . . . . . 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 2927 . . . . . . . 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 6160 . . . . . . . . . . . 12 (Fun 𝑔𝑔:dom 𝑔onto→ran 𝑔)
48 fof 6153 . . . . . . . . . . . 12 (𝑔:dom 𝑔onto→ran 𝑔𝑔:dom 𝑔⟶ran 𝑔)
4947, 48sylbi 207 . . . . . . . . . . 11 (Fun 𝑔𝑔:dom 𝑔⟶ran 𝑔)
5049ad2antrl 764 . . . . . . . . . 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 6399 . . . . . . . . 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 2718 . . . . . . . . 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 2837 . . . . . . 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 2651 . . . . . 6 𝒫 (𝑔𝑥) = 𝒫 (𝑔𝑥)
57 eqid 2651 . . . . . 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 2651 . . . . . 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 simprl 809 . . . . . . . . 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)
60 funfn 5956 . . . . . . . . 9 (Fun 𝑔𝑔 Fn dom 𝑔)
6159, 60sylib 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)
62 ssun1 3809 . . . . . . . . . 10 (𝑔𝑘) ⊆ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
63 fvex 6239 . . . . . . . . . . 11 (𝑔𝑘) ∈ V
6463elpw 4197 . . . . . . . . . 10 ((𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ↔ (𝑔𝑘) ⊆ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
6562, 64mpbir 221 . . . . . . . . 9 (𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
6665rgenw 2953 . . . . . . . 8 𝑘 ∈ dom 𝑔(𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
6739elixp 7957 . . . . . . . 8 (𝑔X𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ↔ (𝑔 Fn dom 𝑔 ∧ ∀𝑘 ∈ dom 𝑔(𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})))
6861, 66, 67sylanblrc 698 . . . . . . 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)}))
69 simpl 472 . . . . . . . 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))))
70 snex 4938 . . . . . . . . . . . . 13 {𝒫 (𝑔𝑥)} ∈ V
7142, 70unex 6998 . . . . . . . . . . . 12 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∈ V
72 ssun2 3810 . . . . . . . . . . . . 13 {𝒫 (𝑔𝑥)} ⊆ ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})
7342uniex 6995 . . . . . . . . . . . . . . 15 (𝑔𝑥) ∈ V
7473pwex 4878 . . . . . . . . . . . . . 14 𝒫 (𝑔𝑥) ∈ V
7574snid 4241 . . . . . . . . . . . . 13 𝒫 (𝑔𝑥) ∈ {𝒫 (𝑔𝑥)}
7672, 75sselii 3633 . . . . . . . . . . . 12 𝒫 (𝑔𝑥) ∈ ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})
77 epttop 20861 . . . . . . . . . . . 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)})))
7871, 76, 77mp2an 708 . . . . . . . . . . 11 {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} ∈ (TopOn‘((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))
7978topontopi 20768 . . . . . . . . . 10 {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} ∈ Top
8079a1i 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)
81 eqid 2651 . . . . . . . . 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)}))})
8280, 81fmptd 6425 . . . . . . . 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)
8340mptex 6527 . . . . . . . . 9 (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) ∈ V
84 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)}))}))
85 dmeq 5356 . . . . . . . . . . . 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)}))}))
8671pwex 4878 . . . . . . . . . . . . . 14 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∈ V
8786rabex 4845 . . . . . . . . . . . . 13 {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} ∈ V
8887, 81dmmpti 6061 . . . . . . . . . . . 12 dom (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) = dom 𝑔
8985, 88syl6eq 2701 . . . . . . . . . . 11 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → dom 𝑓 = dom 𝑔)
9084, 89feq12d 6071 . . . . . . . . . 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))
9189ixpeq1d 7962 . . . . . . . . . . . 12 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) = X𝑘 ∈ dom 𝑔𝒫 (𝑓𝑘))
92 fveq1 6228 . . . . . . . . . . . . . . . . 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))
93 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑘 → (𝑔𝑥) = (𝑔𝑘))
9493unieqd 4478 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘 (𝑔𝑥) = (𝑔𝑘))
9594pweqd 4196 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑘 → 𝒫 (𝑔𝑥) = 𝒫 (𝑔𝑘))
9695sneqd 4222 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑘 → {𝒫 (𝑔𝑥)} = {𝒫 (𝑔𝑘)})
9793, 96uneq12d 3801 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
9897pweqd 4196 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑘 → 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) = 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
9995eleq1d 2715 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → (𝒫 (𝑔𝑥) ∈ 𝑦 ↔ 𝒫 (𝑔𝑘) ∈ 𝑦))
10097eqeq2d 2661 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → (𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ↔ 𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})))
10199, 100imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑘 → ((𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})) ↔ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))))
10298, 101rabeqbidv 3226 . . . . . . . . . . . . . . . . . 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)}))})
103 snex 4938 . . . . . . . . . . . . . . . . . . . . 21 {𝒫 (𝑔𝑘)} ∈ V
10463, 103unex 6998 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∈ V
105104pwex 4878 . . . . . . . . . . . . . . . . . . 19 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∈ V
106105rabex 4845 . . . . . . . . . . . . . . . . . 18 {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))} ∈ V
107102, 81, 106fvmpt 6321 . . . . . . . . . . . . . . . . 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)}))})
10892, 107sylan9eq 2705 . . . . . . . . . . . . . . . 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)}))})
109108unieqd 4478 . . . . . . . . . . . . . . 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)}))})
110 ssun2 3810 . . . . . . . . . . . . . . . . . 18 {𝒫 (𝑔𝑘)} ⊆ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
11163uniex 6995 . . . . . . . . . . . . . . . . . . . 20 (𝑔𝑘) ∈ V
112111pwex 4878 . . . . . . . . . . . . . . . . . . 19 𝒫 (𝑔𝑘) ∈ V
113112snid 4241 . . . . . . . . . . . . . . . . . 18 𝒫 (𝑔𝑘) ∈ {𝒫 (𝑔𝑘)}
114110, 113sselii 3633 . . . . . . . . . . . . . . . . 17 𝒫 (𝑔𝑘) ∈ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
115 epttop 20861 . . . . . . . . . . . . . . . . 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)})))
116104, 114, 115mp2an 708 . . . . . . . . . . . . . . . 16 {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))} ∈ (TopOn‘((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
117116toponunii 20769 . . . . . . . . . . . . . . 15 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) = {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))}
118109, 117syl6eqr 2703 . . . . . . . . . . . . . 14 ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → (𝑓𝑘) = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
119118pweqd 4196 . . . . . . . . . . . . 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)}))
120119ixpeq2dva 7965 . . . . . . . . . . . 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)}))
12191, 120eqtrd 2685 . . . . . . . . . . 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)}))
122 fveq2 6229 . . . . . . . . . . . . . 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)}))})))
123122fveq2d 6233 . . . . . . . . . . . . 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)}))}))))
12489ixpeq1d 7962 . . . . . . . . . . . . 13 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → X𝑘 ∈ dom 𝑓(𝑠𝑘) = X𝑘 ∈ dom 𝑔(𝑠𝑘))
125123, 124fveq12d 6235 . . . . . . . . . . . 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)))
12689ixpeq1d 7962 . . . . . . . . . . . . 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)))
127108fveq2d 6233 . . . . . . . . . . . . . . 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)}))}))
128127fveq1d 6231 . . . . . . . . . . . . . 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)))
129128ixpeq2dva 7965 . . . . . . . . . . . . 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)))
130126, 129eqtrd 2685 . . . . . . . . . . . 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)))
131125, 130eqeq12d 2666 . . . . . . . . . . 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))))
132121, 131raleqbidv 3182 . . . . . . . . . 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))))
13390, 132imbi12d 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)))))
13483, 133spcv 3330 . . . . . . . 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))))
13569, 82, 134sylc 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)))
136 fveq1 6228 . . . . . . . . . . . 12 (𝑠 = 𝑔 → (𝑠𝑘) = (𝑔𝑘))
137136ixpeq2dv 7966 . . . . . . . . . . 11 (𝑠 = 𝑔X𝑘 ∈ dom 𝑔(𝑠𝑘) = X𝑘 ∈ dom 𝑔(𝑔𝑘))
138 fveq2 6229 . . . . . . . . . . . 12 (𝑘 = 𝑥 → (𝑔𝑘) = (𝑔𝑥))
139138cbvixpv 7968 . . . . . . . . . . 11 X𝑘 ∈ dom 𝑔(𝑔𝑘) = X𝑥 ∈ dom 𝑔(𝑔𝑥)
140137, 139syl6eq 2701 . . . . . . . . . 10 (𝑠 = 𝑔X𝑘 ∈ dom 𝑔(𝑠𝑘) = X𝑥 ∈ dom 𝑔(𝑔𝑥))
141140fveq2d 6233 . . . . . . . . 9 (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)))
142136fveq2d 6233 . . . . . . . . . . 11 (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)))
143142ixpeq2dv 7966 . . . . . . . . . 10 (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)))
144138unieqd 4478 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 (𝑔𝑘) = (𝑔𝑥))
145144pweqd 4196 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → 𝒫 (𝑔𝑘) = 𝒫 (𝑔𝑥))
146145sneqd 4222 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑥 → {𝒫 (𝑔𝑘)} = {𝒫 (𝑔𝑥)})
147138, 146uneq12d 3801 . . . . . . . . . . . . . . 15 (𝑘 = 𝑥 → ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))
148147pweqd 4196 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) = 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))
149145eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑘 = 𝑥 → (𝒫 (𝑔𝑘) ∈ 𝑦 ↔ 𝒫 (𝑔𝑥) ∈ 𝑦))
150147eqeq2d 2661 . . . . . . . . . . . . . . 15 (𝑘 = 𝑥 → (𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ↔ 𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})))
151149, 150imbi12d 333 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → ((𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})) ↔ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))))
152148, 151rabeqbidv 3226 . . . . . . . . . . . . 13 (𝑘 = 𝑥 → {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))} = {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))})
153152fveq2d 6233 . . . . . . . . . . . 12 (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)}))}))
154153, 138fveq12d 6235 . . . . . . . . . . 11 (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)))
155154cbvixpv 7968 . . . . . . . . . 10 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))
156143, 155syl6eq 2701 . . . . . . . . 9 (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)))
157141, 156eqeq12d 2666 . . . . . . . 8 (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))))
158157rspcv 3336 . . . . . . 7 (𝑔X𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) → (∀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)) → ((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))))
15968, 135, 158sylc 65 . . . . . 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)))
16041, 43, 55, 56, 57, 58, 159dfac14lem 21468 . . . . 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) ≠ ∅)
161160ex 449 . . . 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) ≠ ∅))
162161alrimiv 1895 . . 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) ≠ ∅))
163 dfac9 8996 . . 3 (CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
164162, 163sylibr 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)
16538, 164impbii 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 383  wal 1521   = wceq 1523  wcel 2030  wne 2823  wnel 2926  wral 2941  {crab 2945  Vcvv 3231  cun 3605  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468   ciun 4552  cmpt 4762  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  Xcixp 7950  AC wacn 8802  CHOICEwac 8976  tcpt 16146  Topctop 20746  TopOnctopon 20763  clsccl 20870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-fin 8001  df-fi 8358  df-card 8803  df-acn 8806  df-ac 8977  df-topgen 16151  df-pt 16152  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator