Theorem qtopres 22757

Description: The quotient topology is unaffected by restriction to the base set. This property makes it slightly more convenient to use, since we don't have to require that 𝐹 be a function with domain 𝑋. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypothesis

Ref	Expression
qtopval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
qtopres	⊢ (𝐹 ∈ 𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))

Proof of Theorem qtopres

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

Step	Hyp	Ref	Expression
1		resima 5914	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑋) “ 𝑋) = (𝐹 “ 𝑋)
2	1	pweqi 4548	. . . . . 6 ⊢ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) = 𝒫 (𝐹 “ 𝑋)
3	2	rabeqi 3406	. . . . 5 ⊢ {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
4		residm 5913	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑋) ↾ 𝑋) = (𝐹 ↾ 𝑋)
5	4	cnveqi 5772	. . . . . . . . 9 ⊢ ◡((𝐹 ↾ 𝑋) ↾ 𝑋) = ◡(𝐹 ↾ 𝑋)
6	5	imaeq1i 5955	. . . . . . . 8 ⊢ (◡((𝐹 ↾ 𝑋) ↾ 𝑋) “ 𝑠) = (◡(𝐹 ↾ 𝑋) “ 𝑠)
7		cnvresima 6122	. . . . . . . 8 ⊢ (◡((𝐹 ↾ 𝑋) ↾ 𝑋) “ 𝑠) = ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋)
8		cnvresima 6122	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝑋) “ 𝑠) = ((◡𝐹 “ 𝑠) ∩ 𝑋)
9	6, 7, 8	3eqtr3i 2774	. . . . . . 7 ⊢ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) = ((◡𝐹 “ 𝑠) ∩ 𝑋)
10	9	eleq1i 2829	. . . . . 6 ⊢ (((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽)
11	10	rabbii 3397	. . . . 5 ⊢ {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽}
12	3, 11	eqtr2i 2767	. . . 4 ⊢ {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
13		qtopval.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
14	13	qtopval 22754	. . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
15		resexg 5926	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝑋) ∈ V)
16	13	qtopval 22754	. . . . 5 ⊢ ((𝐽 ∈ V ∧ (𝐹 ↾ 𝑋) ∈ V) → (𝐽 qTop (𝐹 ↾ 𝑋)) = {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
17	15, 16	sylan2 592	. . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop (𝐹 ↾ 𝑋)) = {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
18	12, 14, 17	3eqtr4a 2805	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))
19	18	expcom 413	. 2 ⊢ (𝐹 ∈ 𝑉 → (𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋))))
20		df-qtop 17135	. . . . 5 ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
21	20	reldmmpo 7386	. . . 4 ⊢ Rel dom qTop
22	21	ovprc1 7294	. . 3 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop 𝐹) = ∅)
23	21	ovprc1 7294	. . 3 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop (𝐹 ↾ 𝑋)) = ∅)
24	22, 23	eqtr4d 2781	. 2 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))
25	19, 24	pm2.61d1 180	1 ⊢ (𝐹 ∈ 𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422   ∩ cin 3882  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  ^◡ccnv 5579   ↾ cres 5582   “ cima 5583  (class class class)co 7255   qTop cqtop 17131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-qtop 17135

This theorem is referenced by:  qtoptop2  22758