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Theorem dfac14 21340
Description: Theorem ptcls 21338 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 6153 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝑓𝑘) = (𝑓𝑥))
21unieqd 4417 . . . . . . . . 9 (𝑘 = 𝑥 (𝑓𝑘) = (𝑓𝑥))
32pweqd 4140 . . . . . . . 8 (𝑘 = 𝑥 → 𝒫 (𝑓𝑘) = 𝒫 (𝑓𝑥))
43cbvixpv 7877 . . . . . . 7 X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) = X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)
54eleq2i 2690 . . . . . 6 (𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) ↔ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥))
6 simplr 791 . . . . . . . . . . 11 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑓:dom 𝑓⟶Top)
76feqmptd 6211 . . . . . . . . . 10 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑓 = (𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘)))
87fveq2d 6157 . . . . . . . . 9 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → (∏t𝑓) = (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘))))
98fveq2d 6157 . . . . . . . 8 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → (cls‘(∏t𝑓)) = (cls‘(∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘)))))
109fveq1d 6155 . . . . . . 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 2621 . . . . . . . 8 (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘))) = (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓𝑘)))
12 vex 3192 . . . . . . . . . 10 𝑓 ∈ V
1312dmex 7053 . . . . . . . . 9 dom 𝑓 ∈ V
1413a1i 11 . . . . . . . 8 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → dom 𝑓 ∈ V)
156ffvelrnda 6320 . . . . . . . . 9 ((((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑓𝑘) ∈ Top)
16 eqid 2621 . . . . . . . . . 10 (𝑓𝑘) = (𝑓𝑘)
1716toptopon 20653 . . . . . . . . 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 477 . . . . . . . . . . . 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 3192 . . . . . . . . . . . . 13 𝑠 ∈ V
2221elixp 7866 . . . . . . . . . . . 12 (𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘) ↔ (𝑠 Fn dom 𝑓 ∧ ∀𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ 𝒫 (𝑓𝑘)))
2322simprbi 480 . . . . . . . . . . 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 2927 . . . . . . . . 9 ((((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑠𝑘) ∈ 𝒫 (𝑓𝑘))
2625elpwid 4146 . . . . . . . 8 ((((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑠𝑘) ⊆ (𝑓𝑘))
27 fvex 6163 . . . . . . . . . 10 (𝑠𝑘) ∈ V
2813, 27iunex 7100 . . . . . . . . 9 𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ V
29 simpll 789 . . . . . . . . . 10 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → CHOICE)
30 acacni 8913 . . . . . . . . . 10 ((CHOICE ∧ dom 𝑓 ∈ V) → AC dom 𝑓 = V)
3129, 13, 30sylancl 693 . . . . . . . . 9 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → AC dom 𝑓 = V)
3228, 31syl5eleqr 2705 . . . . . . . 8 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → 𝑘 ∈ dom 𝑓(𝑠𝑘) ∈ AC dom 𝑓)
3311, 14, 18, 26, 32ptclsg 21337 . . . . . . 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 2655 . . . . . 6 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑥 ∈ dom 𝑓𝒫 (𝑓𝑥)) → ((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)))
355, 34sylan2b 492 . . . . 5 (((CHOICE𝑓:dom 𝑓⟶Top) ∧ 𝑠X𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)) → ((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)))
3635ralrimiva 2961 . . . 4 ((CHOICE𝑓:dom 𝑓⟶Top) → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘)))
3736ex 450 . . 3 (CHOICE → (𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))))
3837alrimiv 1852 . 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 3192 . . . . . . . 8 𝑔 ∈ V
4039dmex 7053 . . . . . . 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 6163 . . . . . . 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 800 . . . . . . . 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 2894 . . . . . . . 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 6084 . . . . . . . . . . . 12 (Fun 𝑔𝑔:dom 𝑔onto→ran 𝑔)
48 fof 6077 . . . . . . . . . . . 12 (𝑔:dom 𝑔onto→ran 𝑔𝑔:dom 𝑔⟶ran 𝑔)
4947, 48sylbi 207 . . . . . . . . . . 11 (Fun 𝑔𝑔:dom 𝑔⟶ran 𝑔)
5049ad2antrl 763 . . . . . . . . . 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 6320 . . . . . . . . 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 2686 . . . . . . . . 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 2804 . . . . . . 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 2621 . . . . . 6 𝒫 (𝑔𝑥) = 𝒫 (𝑔𝑥)
57 eqid 2621 . . . . . 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 2621 . . . . . 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 793 . . . . . . . . 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 5882 . . . . . . . . 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 3759 . . . . . . . . . 10 (𝑔𝑘) ⊆ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
63 fvex 6163 . . . . . . . . . . 11 (𝑔𝑘) ∈ V
6463elpw 4141 . . . . . . . . . 10 ((𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ↔ (𝑔𝑘) ⊆ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
6562, 64mpbir 221 . . . . . . . . 9 (𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
6665rgenw 2919 . . . . . . . 8 𝑘 ∈ dom 𝑔(𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
6739elixp 7866 . . . . . . . 8 (𝑔X𝑘 ∈ dom 𝑔𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ↔ (𝑔 Fn dom 𝑔 ∧ ∀𝑘 ∈ dom 𝑔(𝑔𝑘) ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})))
6861, 66, 67sylanblrc 696 . . . . . . 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 473 . . . . . . . 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 4874 . . . . . . . . . . . . 13 {𝒫 (𝑔𝑥)} ∈ V
7142, 70unex 6916 . . . . . . . . . . . 12 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∈ V
72 ssun2 3760 . . . . . . . . . . . . 13 {𝒫 (𝑔𝑥)} ⊆ ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})
7342uniex 6913 . . . . . . . . . . . . . . 15 (𝑔𝑥) ∈ V
7473pwex 4813 . . . . . . . . . . . . . 14 𝒫 (𝑔𝑥) ∈ V
7574snid 4184 . . . . . . . . . . . . 13 𝒫 (𝑔𝑥) ∈ {𝒫 (𝑔𝑥)}
7672, 75sselii 3584 . . . . . . . . . . . 12 𝒫 (𝑔𝑥) ∈ ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})
77 epttop 20732 . . . . . . . . . . . 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 707 . . . . . . . . . . 11 {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} ∈ (TopOn‘((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))
7978topontopi 20651 . . . . . . . . . 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 2621 . . . . . . . . 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 6346 . . . . . . . 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 6446 . . . . . . . . 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 5289 . . . . . . . . . . . 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 4813 . . . . . . . . . . . . . 14 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∈ V
8786rabex 4778 . . . . . . . . . . . . 13 {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))} ∈ V
8887, 81dmmpti 5985 . . . . . . . . . . . 12 dom (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) = dom 𝑔
8985, 88syl6eq 2671 . . . . . . . . . . 11 (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) → dom 𝑓 = dom 𝑔)
9084, 89feq12d 5995 . . . . . . . . . 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 7871 . . . . . . . . . . . 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 6152 . . . . . . . . . . . . . . . . 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 6153 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑘 → (𝑔𝑥) = (𝑔𝑘))
9493unieqd 4417 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘 (𝑔𝑥) = (𝑔𝑘))
9594pweqd 4140 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑘 → 𝒫 (𝑔𝑥) = 𝒫 (𝑔𝑘))
9695sneqd 4165 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑘 → {𝒫 (𝑔𝑥)} = {𝒫 (𝑔𝑘)})
9793, 96uneq12d 3751 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
9897pweqd 4140 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑘 → 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) = 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
9995eleq1d 2683 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → (𝒫 (𝑔𝑥) ∈ 𝑦 ↔ 𝒫 (𝑔𝑘) ∈ 𝑦))
10097eqeq2d 2631 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑘 → (𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ↔ 𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})))
10199, 100imbi12d 334 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑘 → ((𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})) ↔ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))))
10298, 101rabeqbidv 3184 . . . . . . . . . . . . . . . . . 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 4874 . . . . . . . . . . . . . . . . . . . . 21 {𝒫 (𝑔𝑘)} ∈ V
10463, 103unex 6916 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∈ V
105104pwex 4813 . . . . . . . . . . . . . . . . . . 19 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∈ V
106105rabex 4778 . . . . . . . . . . . . . . . . . 18 {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))} ∈ V
107102, 81, 106fvmpt 6244 . . . . . . . . . . . . . . . . 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 2675 . . . . . . . . . . . . . . . 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 4417 . . . . . . . . . . . . . . 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 3760 . . . . . . . . . . . . . . . . . 18 {𝒫 (𝑔𝑘)} ⊆ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
11163uniex 6913 . . . . . . . . . . . . . . . . . . . 20 (𝑔𝑘) ∈ V
112111pwex 4813 . . . . . . . . . . . . . . . . . . 19 𝒫 (𝑔𝑘) ∈ V
113112snid 4184 . . . . . . . . . . . . . . . . . 18 𝒫 (𝑔𝑘) ∈ {𝒫 (𝑔𝑘)}
114110, 113sselii 3584 . . . . . . . . . . . . . . . . 17 𝒫 (𝑔𝑘) ∈ ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})
115 epttop 20732 . . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . 16 {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))} ∈ (TopOn‘((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
117116toponunii 20652 . . . . . . . . . . . . . . 15 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) = {𝑦 ∈ 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ∣ (𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))}
118109, 117syl6eqr 2673 . . . . . . . . . . . . . 14 ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}) ∣ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → (𝑓𝑘) = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}))
119118pweqd 4140 . . . . . . . . . . . . 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 7874 . . . . . . . . . . . 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 2655 . . . . . . . . . . 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 6153 . . . . . . . . . . . . . 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 6157 . . . . . . . . . . . . 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 7871 . . . . . . . . . . . . 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 6159 . . . . . . . . . . . 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 7871 . . . . . . . . . . . . 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 6157 . . . . . . . . . . . . . . 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 6155 . . . . . . . . . . . . . 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 7874 . . . . . . . . . . . . 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 2655 . . . . . . . . . . . 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 2636 . . . . . . . . . . 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 3144 . . . . . . . . . 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 334 . . . . . . . . 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 3288 . . . . . . . 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 6152 . . . . . . . . . . . 12 (𝑠 = 𝑔 → (𝑠𝑘) = (𝑔𝑘))
137136ixpeq2dv 7875 . . . . . . . . . . 11 (𝑠 = 𝑔X𝑘 ∈ dom 𝑔(𝑠𝑘) = X𝑘 ∈ dom 𝑔(𝑔𝑘))
138 fveq2 6153 . . . . . . . . . . . 12 (𝑘 = 𝑥 → (𝑔𝑘) = (𝑔𝑥))
139138cbvixpv 7877 . . . . . . . . . . 11 X𝑘 ∈ dom 𝑔(𝑔𝑘) = X𝑥 ∈ dom 𝑔(𝑔𝑥)
140137, 139syl6eq 2671 . . . . . . . . . 10 (𝑠 = 𝑔X𝑘 ∈ dom 𝑔(𝑠𝑘) = X𝑥 ∈ dom 𝑔(𝑔𝑥))
141140fveq2d 6157 . . . . . . . . 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 6157 . . . . . . . . . . 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 7875 . . . . . . . . . 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 4417 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 (𝑔𝑘) = (𝑔𝑥))
145144pweqd 4140 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → 𝒫 (𝑔𝑘) = 𝒫 (𝑔𝑥))
146145sneqd 4165 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑥 → {𝒫 (𝑔𝑘)} = {𝒫 (𝑔𝑥)})
147138, 146uneq12d 3751 . . . . . . . . . . . . . . 15 (𝑘 = 𝑥 → ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))
148147pweqd 4140 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → 𝒫 ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) = 𝒫 ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))
149145eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑘 = 𝑥 → (𝒫 (𝑔𝑘) ∈ 𝑦 ↔ 𝒫 (𝑔𝑥) ∈ 𝑦))
150147eqeq2d 2631 . . . . . . . . . . . . . . 15 (𝑘 = 𝑥 → (𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)}) ↔ 𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)})))
151149, 150imbi12d 334 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → ((𝒫 (𝑔𝑘) ∈ 𝑦𝑦 = ((𝑔𝑘) ∪ {𝒫 (𝑔𝑘)})) ↔ (𝒫 (𝑔𝑥) ∈ 𝑦𝑦 = ((𝑔𝑥) ∪ {𝒫 (𝑔𝑥)}))))
152148, 151rabeqbidv 3184 . . . . . . . . . . . . 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 6157 . . . . . . . . . . . 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 6159 . . . . . . . . . . 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 7877 . . . . . . . . . 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 2671 . . . . . . . . 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 2636 . . . . . . . 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 3294 . . . . . . 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 21339 . . . . 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 450 . . . 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 1852 . . 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 8909 . . 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 384  wal 1478   = wceq 1480  wcel 1987  wne 2790  wnel 2893  wral 2907  {crab 2911  Vcvv 3189  cun 3557  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407   ciun 4490  cmpt 4678  dom cdm 5079  ran crn 5080  Fun wfun 5846   Fn wfn 5847  wf 5848  ontowfo 5850  cfv 5852  Xcixp 7859  AC wacn 8715  CHOICEwac 8889  tcpt 16027  Topctop 20626  TopOnctopon 20643  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-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  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-se 5039  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-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  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-dom 7908  df-fin 7910  df-fi 8268  df-card 8716  df-acn 8719  df-ac 8890  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: (None)
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