Theorem qtopval 23589

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 3471	. 2 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ V)
2		elex 3471	. 2 ⊢ (𝐹 ∈ 𝑊 → 𝐹 ∈ V)
3		imaexg 7892	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹 “ 𝑋) ∈ V)
4		pwexg 5336	. . . . 5 ⊢ ((𝐹 “ 𝑋) ∈ V → 𝒫 (𝐹 “ 𝑋) ∈ V)
5		rabexg 5295	. . . . 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 4887	. . . . . . . 8 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = ∪ 𝐽)
11		qtopval.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
12	10, 11	eqtr4di 2783	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = 𝑋)
13	8, 12	imaeq12d 6035	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (𝑓 “ ∪ 𝑗) = (𝐹 “ 𝑋))
14	13	pweqd 4583	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝒫 (𝑓 “ ∪ 𝑗) = 𝒫 (𝐹 “ 𝑋))
15	8	cnveqd 5842	. . . . . . . 8 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
16	15	imaeq1d 6033	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (◡𝑓 “ 𝑠) = (◡𝐹 “ 𝑠))
17	16, 12	ineq12d 4187	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) = ((◡𝐹 “ 𝑠) ∩ 𝑋))
18	17, 9	eleq12d 2823	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽))
19	14, 18	rabeqbidv 3427	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
20		df-qtop 17477	. . . 4 ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
21	19, 20	ovmpoga 7546	. . 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 1540   ∈ wcel 2109  {crab 3408  Vcvv 3450   ∩ cin 3916  𝒫 cpw 4566  ∪ cuni 4874  ^◡ccnv 5640   “ cima 5644  (class class class)co 7390   qTop cqtop 17473

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-qtop 17477

This theorem is referenced by:  qtopval2  23590  qtopres  23592  imastopn  23614