Theorem qtopval 23199

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 3493	. 2 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ V)
2		elex 3493	. 2 ⊢ (𝐹 ∈ 𝑊 → 𝐹 ∈ V)
3		imaexg 7906	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹 “ 𝑋) ∈ V)
4		pwexg 5377	. . . . 5 ⊢ ((𝐹 “ 𝑋) ∈ V → 𝒫 (𝐹 “ 𝑋) ∈ V)
5		rabexg 5332	. . . . 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 483	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ V) → {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V)
8		simpr 486	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
9		simpl 484	. . . . . . . . 9 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝑗 = 𝐽)
10	9	unieqd 4923	. . . . . . . 8 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = ∪ 𝐽)
11		qtopval.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
12	10, 11	eqtr4di 2791	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = 𝑋)
13	8, 12	imaeq12d 6061	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (𝑓 “ ∪ 𝑗) = (𝐹 “ 𝑋))
14	13	pweqd 4620	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝒫 (𝑓 “ ∪ 𝑗) = 𝒫 (𝐹 “ 𝑋))
15	8	cnveqd 5876	. . . . . . . 8 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
16	15	imaeq1d 6059	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (◡𝑓 “ 𝑠) = (◡𝐹 “ 𝑠))
17	16, 12	ineq12d 4214	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) = ((◡𝐹 “ 𝑠) ∩ 𝑋))
18	17, 9	eleq12d 2828	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽))
19	14, 18	rabeqbidv 3450	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
20		df-qtop 17453	. . . 4 ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
21	19, 20	ovmpoga 7562	. . 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 1463	. 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 397   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   ∩ cin 3948  𝒫 cpw 4603  ∪ cuni 4909  ^◡ccnv 5676   “ cima 5680  (class class class)co 7409   qTop cqtop 17449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-qtop 17453

This theorem is referenced by:  qtopval2  23200  qtopres  23202  imastopn  23224