Theorem dfac14 23505

Description: Theorem ptcls 23503 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 → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))))

Distinct variable group:   𝑓,𝑘,𝑠

Proof of Theorem dfac14

Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → (𝑓‘𝑘) = (𝑓‘𝑥))
2	1	unieqd 4884	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → ∪ (𝑓‘𝑘) = ∪ (𝑓‘𝑥))
3	2	pweqd 4580	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → 𝒫 ∪ (𝑓‘𝑘) = 𝒫 ∪ (𝑓‘𝑥))
4	3	cbvixpv 8888	. . . . . . 7 ⊢ X𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘) = X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)
5	4	eleq2i 2820	. . . . . 6 ⊢ (𝑠 ∈ X𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘) ↔ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥))
6		simplr 768	. . . . . . . . . . 11 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → 𝑓:dom 𝑓⟶Top)
7	6	feqmptd 6929	. . . . . . . . . 10 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → 𝑓 = (𝑘 ∈ dom 𝑓 ↦ (𝑓‘𝑘)))
8	7	fveq2d 6862	. . . . . . . . 9 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → (∏t‘𝑓) = (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓‘𝑘))))
9	8	fveq2d 6862	. . . . . . . 8 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → (cls‘(∏t‘𝑓)) = (cls‘(∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓‘𝑘)))))
10	9	fveq1d 6860	. . . . . . 7 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → ((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = ((cls‘(∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓‘𝑘))))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)))
11		eqid 2729	. . . . . . . 8 ⊢ (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓‘𝑘))) = (∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓‘𝑘)))
12		vex 3451	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
13	12	dmex 7885	. . . . . . . . 9 ⊢ dom 𝑓 ∈ V
14	13	a1i 11	. . . . . . . 8 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → dom 𝑓 ∈ V)
15	6	ffvelcdmda 7056	. . . . . . . . 9 ⊢ ((((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑓‘𝑘) ∈ Top)
16		toptopon2 22805	. . . . . . . . 9 ⊢ ((𝑓‘𝑘) ∈ Top ↔ (𝑓‘𝑘) ∈ (TopOn‘∪ (𝑓‘𝑘)))
17	15, 16	sylib 218	. . . . . . . 8 ⊢ ((((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑓‘𝑘) ∈ (TopOn‘∪ (𝑓‘𝑘)))
18		simpr 484	. . . . . . . . . . . 12 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥))
19	18, 5	sylibr 234	. . . . . . . . . . 11 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → 𝑠 ∈ X𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘))
20		vex 3451	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
21	20	elixp 8877	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ X𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘) ↔ (𝑠 Fn dom 𝑓 ∧ ∀𝑘 ∈ dom 𝑓(𝑠‘𝑘) ∈ 𝒫 ∪ (𝑓‘𝑘)))
22	21	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑠 ∈ X𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘) → ∀𝑘 ∈ dom 𝑓(𝑠‘𝑘) ∈ 𝒫 ∪ (𝑓‘𝑘))
23	19, 22	syl 17	. . . . . . . . . 10 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → ∀𝑘 ∈ dom 𝑓(𝑠‘𝑘) ∈ 𝒫 ∪ (𝑓‘𝑘))
24	23	r19.21bi 3229	. . . . . . . . 9 ⊢ ((((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑠‘𝑘) ∈ 𝒫 ∪ (𝑓‘𝑘))
25	24	elpwid 4572	. . . . . . . 8 ⊢ ((((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) ∧ 𝑘 ∈ dom 𝑓) → (𝑠‘𝑘) ⊆ ∪ (𝑓‘𝑘))
26		fvex 6871	. . . . . . . . . 10 ⊢ (𝑠‘𝑘) ∈ V
27	13, 26	iunex 7947	. . . . . . . . 9 ⊢ ∪ 𝑘 ∈ dom 𝑓(𝑠‘𝑘) ∈ V
28		simpll 766	. . . . . . . . . 10 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → CHOICE)
29		acacni 10094	. . . . . . . . . 10 ⊢ ((CHOICE ∧ dom 𝑓 ∈ V) → AC dom 𝑓 = V)
30	28, 13, 29	sylancl 586	. . . . . . . . 9 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → AC dom 𝑓 = V)
31	27, 30	eleqtrrid 2835	. . . . . . . 8 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → ∪ 𝑘 ∈ dom 𝑓(𝑠‘𝑘) ∈ AC dom 𝑓)
32	11, 14, 17, 25, 31	ptclsg 23502	. . . . . . 7 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → ((cls‘(∏t‘(𝑘 ∈ dom 𝑓 ↦ (𝑓‘𝑘))))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)))
33	10, 32	eqtrd 2764	. . . . . 6 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑥 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑥)) → ((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)))
34	5, 33	sylan2b 594	. . . . 5 ⊢ (((CHOICE ∧ 𝑓:dom 𝑓⟶Top) ∧ 𝑠 ∈ X𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)) → ((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)))
35	34	ralrimiva 3125	. . . 4 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Top) → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)))
36	35	ex 412	. . 3 ⊢ (CHOICE → (𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))))
37	36	alrimiv 1927	. 2 ⊢ (CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))))
38		vex 3451	. . . . . . . 8 ⊢ 𝑔 ∈ V
39	38	dmex 7885	. . . . . . 7 ⊢ dom 𝑔 ∈ V
40	39	a1i 11	. . . . . 6 ⊢ ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → dom 𝑔 ∈ V)
41		fvex 6871	. . . . . . 7 ⊢ (𝑔‘𝑥) ∈ V
42	41	a1i 11	. . . . . 6 ⊢ (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ V)
43		simplrr 777	. . . . . . . 8 ⊢ (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → ∅ ∉ ran 𝑔)
44		df-nel 3030	. . . . . . . 8 ⊢ (∅ ∉ ran 𝑔 ↔ ¬ ∅ ∈ ran 𝑔)
45	43, 44	sylib 218	. . . . . . 7 ⊢ (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔)
46		funforn 6779	. . . . . . . . . . . 12 ⊢ (Fun 𝑔 ↔ 𝑔:dom 𝑔–onto→ran 𝑔)
47		fof 6772	. . . . . . . . . . . 12 ⊢ (𝑔:dom 𝑔–onto→ran 𝑔 → 𝑔:dom 𝑔⟶ran 𝑔)
48	46, 47	sylbi 217	. . . . . . . . . . 11 ⊢ (Fun 𝑔 → 𝑔:dom 𝑔⟶ran 𝑔)
49	48	ad2antrl 728	. . . . . . . . . 10 ⊢ ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → 𝑔:dom 𝑔⟶ran 𝑔)
50	49	ffvelcdmda 7056	. . . . . . . . 9 ⊢ (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔)
51		eleq1 2816	. . . . . . . . 9 ⊢ ((𝑔‘𝑥) = ∅ → ((𝑔‘𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔))
52	50, 51	syl5ibcom 245	. . . . . . . 8 ⊢ (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔‘𝑥) = ∅ → ∅ ∈ ran 𝑔))
53	52	necon3bd 2939	. . . . . . 7 ⊢ (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔‘𝑥) ≠ ∅))
54	45, 53	mpd 15	. . . . . 6 ⊢ (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅)
55		eqid 2729	. . . . . 6 ⊢ 𝒫 ∪ (𝑔‘𝑥) = 𝒫 ∪ (𝑔‘𝑥)
56		eqid 2729	. . . . . 6 ⊢ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))} = {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}
57		eqid 2729	. . . . . 6 ⊢ (∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}))
58		fveq1 6857	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑔 → (𝑠‘𝑘) = (𝑔‘𝑘))
59	58	ixpeq2dv 8886	. . . . . . . . . 10 ⊢ (𝑠 = 𝑔 → X𝑘 ∈ dom 𝑔(𝑠‘𝑘) = X𝑘 ∈ dom 𝑔(𝑔‘𝑘))
60		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → (𝑔‘𝑘) = (𝑔‘𝑥))
61	60	cbvixpv 8888	. . . . . . . . . 10 ⊢ X𝑘 ∈ dom 𝑔(𝑔‘𝑘) = X𝑥 ∈ dom 𝑔(𝑔‘𝑥)
62	59, 61	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝑠 = 𝑔 → X𝑘 ∈ dom 𝑔(𝑠‘𝑘) = X𝑥 ∈ dom 𝑔(𝑔‘𝑥))
63	62	fveq2d 6862	. . . . . . . 8 ⊢ (𝑠 = 𝑔 → ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠‘𝑘)) = ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑥 ∈ dom 𝑔(𝑔‘𝑥)))
64	58	fveq2d 6862	. . . . . . . . . 10 ⊢ (𝑠 = 𝑔 → ((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘)) = ((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑔‘𝑘)))
65	64	ixpeq2dv 8886	. . . . . . . . 9 ⊢ (𝑠 = 𝑔 → X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑔‘𝑘)))
66	60	unieqd 4884	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → ∪ (𝑔‘𝑘) = ∪ (𝑔‘𝑥))
67	66	pweqd 4580	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑥 → 𝒫 ∪ (𝑔‘𝑘) = 𝒫 ∪ (𝑔‘𝑥))
68	67	sneqd 4601	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑥 → {𝒫 ∪ (𝑔‘𝑘)} = {𝒫 ∪ (𝑔‘𝑥)})
69	60, 68	uneq12d 4132	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑥 → ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))
70	69	pweqd 4580	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑥 → 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) = 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))
71	67	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑥 → (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 ↔ 𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦))
72	69	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑥 → (𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ↔ 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)})))
73	71, 72	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑥 → ((𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})) ↔ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))))
74	70, 73	rabeqbidv 3424	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑥 → {𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))} = {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})
75	74	fveq2d 6862	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → (cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))}) = (cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}))
76	75, 60	fveq12d 6865	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → ((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑔‘𝑘)) = ((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})‘(𝑔‘𝑥)))
77	76	cbvixpv 8888	. . . . . . . . 9 ⊢ X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑔‘𝑘)) = X𝑥 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})‘(𝑔‘𝑥))
78	65, 77	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑠 = 𝑔 → X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘)) = X𝑥 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})‘(𝑔‘𝑥)))
79	63, 78	eqeq12d 2745	. . . . . . 7 ⊢ (𝑠 = 𝑔 → (((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘)) ↔ ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑥 ∈ dom 𝑔(𝑔‘𝑥)) = X𝑥 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})‘(𝑔‘𝑥))))
80		simpl 482	. . . . . . . 8 ⊢ ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → ∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))))
81		snex 5391	. . . . . . . . . . . . 13 ⊢ {𝒫 ∪ (𝑔‘𝑥)} ∈ V
82	41, 81	unex 7720	. . . . . . . . . . . 12 ⊢ ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∈ V
83		ssun2 4142	. . . . . . . . . . . . 13 ⊢ {𝒫 ∪ (𝑔‘𝑥)} ⊆ ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)})
84	41	uniex 7717	. . . . . . . . . . . . . . 15 ⊢ ∪ (𝑔‘𝑥) ∈ V
85	84	pwex 5335	. . . . . . . . . . . . . 14 ⊢ 𝒫 ∪ (𝑔‘𝑥) ∈ V
86	85	snid 4626	. . . . . . . . . . . . 13 ⊢ 𝒫 ∪ (𝑔‘𝑥) ∈ {𝒫 ∪ (𝑔‘𝑥)}
87	83, 86	sselii 3943	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ (𝑔‘𝑥) ∈ ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)})
88		epttop 22896	. . . . . . . . . . . 12 ⊢ ((((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∈ V ∧ 𝒫 ∪ (𝑔‘𝑥) ∈ ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)})) → {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))} ∈ (TopOn‘((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)})))
89	82, 87, 88	mp2an 692	. . . . . . . . . . 11 ⊢ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))} ∈ (TopOn‘((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))
90	89	topontopi 22802	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))} ∈ Top
91	90	a1i 11	. . . . . . . . 9 ⊢ (((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) ∧ 𝑥 ∈ dom 𝑔) → {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))} ∈ Top)
92	91	fmpttd 7087	. . . . . . . 8 ⊢ ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}):dom 𝑔⟶Top)
93	39	mptex 7197	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) ∈ V
94		id 22	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → 𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}))
95		dmeq 5867	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → dom 𝑓 = dom (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}))
96	82	pwex 5335	. . . . . . . . . . . . . 14 ⊢ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∈ V
97	96	rabex 5294	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))} ∈ V
98		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})
99	97, 98	dmmpti 6662	. . . . . . . . . . . 12 ⊢ dom (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) = dom 𝑔
100	95, 99	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → dom 𝑓 = dom 𝑔)
101	94, 100	feq12d 6676	. . . . . . . . . 10 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → (𝑓:dom 𝑓⟶Top ↔ (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}):dom 𝑔⟶Top))
102	100	ixpeq1d 8882	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → X𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘) = X𝑘 ∈ dom 𝑔𝒫 ∪ (𝑓‘𝑘))
103		fveq1 6857	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → (𝑓‘𝑘) = ((𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})‘𝑘))
104		fveq2 6858	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑘 → (𝑔‘𝑥) = (𝑔‘𝑘))
105	104	unieqd 4884	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑘 → ∪ (𝑔‘𝑥) = ∪ (𝑔‘𝑘))
106	105	pweqd 4580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑘 → 𝒫 ∪ (𝑔‘𝑥) = 𝒫 ∪ (𝑔‘𝑘))
107	106	sneqd 4601	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑘 → {𝒫 ∪ (𝑔‘𝑥)} = {𝒫 ∪ (𝑔‘𝑘)})
108	104, 107	uneq12d 4132	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑘 → ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))
109	108	pweqd 4580	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) = 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))
110	106	eleq1d 2813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑘 → (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 ↔ 𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦))
111	108	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑘 → (𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ↔ 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})))
112	110, 111	imbi12d 344	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → ((𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)})) ↔ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))))
113	109, 112	rabeqbidv 3424	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑘 → {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))} = {𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})
114		fvex 6871	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔‘𝑘) ∈ V
115		snex 5391	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝒫 ∪ (𝑔‘𝑘)} ∈ V
116	114, 115	unex 7720	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∈ V
117	116	pwex 5335	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∈ V
118	117	rabex 5294	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))} ∈ V
119	113, 98, 118	fvmpt 6968	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ dom 𝑔 → ((𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})‘𝑘) = {𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})
120	103, 119	sylan9eq 2784	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → (𝑓‘𝑘) = {𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})
121	120	unieqd 4884	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → ∪ (𝑓‘𝑘) = ∪ {𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})
122		ssun2 4142	. . . . . . . . . . . . . . . . . 18 ⊢ {𝒫 ∪ (𝑔‘𝑘)} ⊆ ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})
123	114	uniex 7717	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ (𝑔‘𝑘) ∈ V
124	123	pwex 5335	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝒫 ∪ (𝑔‘𝑘) ∈ V
125	124	snid 4626	. . . . . . . . . . . . . . . . . 18 ⊢ 𝒫 ∪ (𝑔‘𝑘) ∈ {𝒫 ∪ (𝑔‘𝑘)}
126	122, 125	sselii 3943	. . . . . . . . . . . . . . . . 17 ⊢ 𝒫 ∪ (𝑔‘𝑘) ∈ ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})
127		epttop 22896	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∈ V ∧ 𝒫 ∪ (𝑔‘𝑘) ∈ ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})) → {𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))} ∈ (TopOn‘((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})))
128	116, 126, 127	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ {𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))} ∈ (TopOn‘((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))
129	128	toponunii 22803	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) = ∪ {𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))}
130	121, 129	eqtr4di 2782	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → ∪ (𝑓‘𝑘) = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))
131	130	pweqd 4580	. . . . . . . . . . . . 13 ⊢ ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → 𝒫 ∪ (𝑓‘𝑘) = 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))
132	131	ixpeq2dva 8885	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → X𝑘 ∈ dom 𝑔𝒫 ∪ (𝑓‘𝑘) = X𝑘 ∈ dom 𝑔𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))
133	102, 132	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → X𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘) = X𝑘 ∈ dom 𝑔𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))
134		2fveq3 6863	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → (cls‘(∏t‘𝑓)) = (cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}))))
135	100	ixpeq1d 8882	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → X𝑘 ∈ dom 𝑓(𝑠‘𝑘) = X𝑘 ∈ dom 𝑔(𝑠‘𝑘))
136	134, 135	fveq12d 6865	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → ((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠‘𝑘)))
137	100	ixpeq1d 8882	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑔((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)))
138	120	fveq2d 6862	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → (cls‘(𝑓‘𝑘)) = (cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))}))
139	138	fveq1d 6860	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) ∧ 𝑘 ∈ dom 𝑔) → ((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)) = ((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘)))
140	139	ixpeq2dva 8885	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → X𝑘 ∈ dom 𝑔((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘)))
141	137, 140	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘)))
142	136, 141	eqeq12d 2745	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → (((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)) ↔ ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘))))
143	133, 142	raleqbidv 3319	. . . . . . . . . 10 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → (∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘)) ↔ ∀𝑠 ∈ X 𝑘 ∈ dom 𝑔𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘))))
144	101, 143	imbi12d 344	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}) → ((𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ↔ ((𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}):dom 𝑔⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑔𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘)))))
145	93, 144	spcv 3571	. . . . . . . 8 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) → ((𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))}):dom 𝑔⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑔𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘))))
146	80, 92, 145	sylc 65	. . . . . . 7 ⊢ ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑔𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑘 ∈ dom 𝑔(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ∣ (𝒫 ∪ (𝑔‘𝑘) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))})‘(𝑠‘𝑘)))
147		simprl 770	. . . . . . . . 9 ⊢ ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → Fun 𝑔)
148	147	funfnd 6547	. . . . . . . 8 ⊢ ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → 𝑔 Fn dom 𝑔)
149		ssun1 4141	. . . . . . . . . 10 ⊢ (𝑔‘𝑘) ⊆ ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})
150	114	elpw 4567	. . . . . . . . . 10 ⊢ ((𝑔‘𝑘) ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ↔ (𝑔‘𝑘) ⊆ ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))
151	149, 150	mpbir 231	. . . . . . . . 9 ⊢ (𝑔‘𝑘) ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})
152	151	rgenw 3048	. . . . . . . 8 ⊢ ∀𝑘 ∈ dom 𝑔(𝑔‘𝑘) ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})
153	38	elixp 8877	. . . . . . . 8 ⊢ (𝑔 ∈ X𝑘 ∈ dom 𝑔𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}) ↔ (𝑔 Fn dom 𝑔 ∧ ∀𝑘 ∈ dom 𝑔(𝑔‘𝑘) ∈ 𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)})))
154	148, 152, 153	sylanblrc 590	. . . . . . 7 ⊢ ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → 𝑔 ∈ X𝑘 ∈ dom 𝑔𝒫 ((𝑔‘𝑘) ∪ {𝒫 ∪ (𝑔‘𝑘)}))
155	79, 146, 154	rspcdva 3589	. . . . . 6 ⊢ ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → ((cls‘(∏t‘(𝑥 ∈ dom 𝑔 ↦ {𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})))‘X𝑥 ∈ dom 𝑔(𝑔‘𝑥)) = X𝑥 ∈ dom 𝑔((cls‘{𝑦 ∈ 𝒫 ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}) ∣ (𝒫 ∪ (𝑔‘𝑥) ∈ 𝑦 → 𝑦 = ((𝑔‘𝑥) ∪ {𝒫 ∪ (𝑔‘𝑥)}))})‘(𝑔‘𝑥)))
156	40, 42, 54, 55, 56, 57, 155	dfac14lem 23504	. . . . 5 ⊢ ((∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) ∧ (Fun 𝑔 ∧ ∅ ∉ ran 𝑔)) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅)
157	156	ex 412	. . . 4 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) → ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
158	157	alrimiv 1927	. . 3 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
159		dfac9 10090	. . 3 ⊢ (CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
160	158, 159	sylibr 234	. 2 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))) → CHOICE)
161	37, 160	impbii 209	1 ⊢ (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∉ wnel 3029  ∀wral 3044  {crab 3405  Vcvv 3447   ∪ cun 3912   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5188  dom cdm 5638  ran crn 5639  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  –onto→wfo 6509  ‘cfv 6511  Xcixp 8870  AC wacn 9891  CHOICEwac 10068  ∏_tcpt 17401  Topctop 22780  TopOnctopon 22797  clsccl 22905

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-fin 8922  df-fi 9362  df-card 9892  df-acn 9895  df-ac 10069  df-topgen 17406  df-pt 17407  df-top 22781  df-topon 22798  df-bases 22833  df-cld 22906  df-ntr 22907  df-cls 22908

This theorem is referenced by: (None)