Theorem qtopres 23654

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 5982	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑋) “ 𝑋) = (𝐹 “ 𝑋)
2	1	pweqi 4572	. . . . . 6 ⊢ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) = 𝒫 (𝐹 “ 𝑋)
3	2	rabeqi 3414	. . . . 5 ⊢ {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
4		residm 5977	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑋) ↾ 𝑋) = (𝐹 ↾ 𝑋)
5	4	cnveqi 5831	. . . . . . . . 9 ⊢ ◡((𝐹 ↾ 𝑋) ↾ 𝑋) = ◡(𝐹 ↾ 𝑋)
6	5	imaeq1i 6024	. . . . . . . 8 ⊢ (◡((𝐹 ↾ 𝑋) ↾ 𝑋) “ 𝑠) = (◡(𝐹 ↾ 𝑋) “ 𝑠)
7		cnvresima 6196	. . . . . . . 8 ⊢ (◡((𝐹 ↾ 𝑋) ↾ 𝑋) “ 𝑠) = ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋)
8		cnvresima 6196	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝑋) “ 𝑠) = ((◡𝐹 “ 𝑠) ∩ 𝑋)
9	6, 7, 8	3eqtr3i 2768	. . . . . . 7 ⊢ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) = ((◡𝐹 “ 𝑠) ∩ 𝑋)
10	9	eleq1i 2828	. . . . . 6 ⊢ (((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽)
11	10	rabbii 3406	. . . . 5 ⊢ {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽}
12	3, 11	eqtr2i 2761	. . . 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 23651	. . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
15		resexg 5994	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝑋) ∈ V)
16	13	qtopval 23651	. . . . 5 ⊢ ((𝐽 ∈ V ∧ (𝐹 ↾ 𝑋) ∈ V) → (𝐽 qTop (𝐹 ↾ 𝑋)) = {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
17	15, 16	sylan2 594	. . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop (𝐹 ↾ 𝑋)) = {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
18	12, 14, 17	3eqtr4a 2798	. . 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 17440	. . . . 5 ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
21	20	reldmmpo 7502	. . . 4 ⊢ Rel dom qTop
22	21	ovprc1 7407	. . 3 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop 𝐹) = ∅)
23	21	ovprc1 7407	. . 3 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop (𝐹 ↾ 𝑋)) = ∅)
24	22, 23	eqtr4d 2775	. 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 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442   ∩ cin 3902  ∅c0 4287  𝒫 cpw 4556  ∪ cuni 4865  ^◡ccnv 5631   ↾ cres 5634   “ cima 5635  (class class class)co 7368   qTop cqtop 17436

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 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-qtop 17440

This theorem is referenced by:  qtoptop2  23655