Theorem qtopval 23660

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 3450	. 2 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ V)
2		elex 3450	. 2 ⊢ (𝐹 ∈ 𝑊 → 𝐹 ∈ V)
3		imaexg 7864	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹 “ 𝑋) ∈ V)
4		pwexg 5320	. . . . 5 ⊢ ((𝐹 “ 𝑋) ∈ V → 𝒫 (𝐹 “ 𝑋) ∈ V)
5		rabexg 5278	. . . . 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 4863	. . . . . . . 8 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = ∪ 𝐽)
11		qtopval.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
12	10, 11	eqtr4di 2789	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = 𝑋)
13	8, 12	imaeq12d 6026	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (𝑓 “ ∪ 𝑗) = (𝐹 “ 𝑋))
14	13	pweqd 4558	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝒫 (𝑓 “ ∪ 𝑗) = 𝒫 (𝐹 “ 𝑋))
15	8	cnveqd 5830	. . . . . . . 8 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
16	15	imaeq1d 6024	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (◡𝑓 “ 𝑠) = (◡𝐹 “ 𝑠))
17	16, 12	ineq12d 4161	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) = ((◡𝐹 “ 𝑠) ∩ 𝑋))
18	17, 9	eleq12d 2830	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽))
19	14, 18	rabeqbidv 3407	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
20		df-qtop 17471	. . . 4 ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
21	19, 20	ovmpoga 7521	. . 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 1465	. 2 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
23	1, 2, 22	syl2an 597	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389  Vcvv 3429   ∩ cin 3888  𝒫 cpw 4541  ∪ cuni 4850  ^◡ccnv 5630   “ cima 5634  (class class class)co 7367   qTop cqtop 17467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-qtop 17471

This theorem is referenced by:  qtopval2  23661  qtopres  23663  imastopn  23685