Theorem qtopval 23611

Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypothesis

Ref	Expression
qtopval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
qtopval	⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})

Distinct variable groups:   𝐹,𝑠   𝐽,𝑠   𝑉,𝑠   𝑋,𝑠

Allowed substitution hint:   𝑊(𝑠)

Proof of Theorem qtopval

Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3457	. 2 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ V)
2		elex 3457	. 2 ⊢ (𝐹 ∈ 𝑊 → 𝐹 ∈ V)
3		imaexg 7843	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹 “ 𝑋) ∈ V)
4		pwexg 5316	. . . . 5 ⊢ ((𝐹 “ 𝑋) ∈ V → 𝒫 (𝐹 “ 𝑋) ∈ V)
5		rabexg 5275	. . . . 5 ⊢ (𝒫 (𝐹 “ 𝑋) ∈ V → {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V)
6	3, 4, 5	3syl 18	. . . 4 ⊢ (𝐹 ∈ V → {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V)
7	6	adantl 481	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ V) → {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V)
8		simpr 484	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
9		simpl 482	. . . . . . . . 9 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝑗 = 𝐽)
10	9	unieqd 4872	. . . . . . . 8 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = ∪ 𝐽)
11		qtopval.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
12	10, 11	eqtr4di 2784	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = 𝑋)
13	8, 12	imaeq12d 6010	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (𝑓 “ ∪ 𝑗) = (𝐹 “ 𝑋))
14	13	pweqd 4567	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝒫 (𝑓 “ ∪ 𝑗) = 𝒫 (𝐹 “ 𝑋))
15	8	cnveqd 5815	. . . . . . . 8 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
16	15	imaeq1d 6008	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (◡𝑓 “ 𝑠) = (◡𝐹 “ 𝑠))
17	16, 12	ineq12d 4171	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) = ((◡𝐹 “ 𝑠) ∩ 𝑋))
18	17, 9	eleq12d 2825	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽))
19	14, 18	rabeqbidv 3413	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
20		df-qtop 17411	. . . 4 ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
21	19, 20	ovmpoga 7500	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ V ∧ {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
22	7, 21	mpd3an3 1464	. 2 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
23	1, 2, 22	syl2an 596	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ∩ cin 3901  𝒫 cpw 4550  ∪ cuni 4859  ^◡ccnv 5615   “ cima 5619  (class class class)co 7346   qTop cqtop 17407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-qtop 17411

This theorem is referenced by:  qtopval2  23612  qtopres  23614  imastopn  23636