Theorem qtopres 23925

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 6008	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑋) “ 𝑋) = (𝐹 “ 𝑋)
2	1	pweqi 4573	. . . . . 6 ⊢ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) = 𝒫 (𝐹 “ 𝑋)
3	2	rabeqi 3425	. . . . 5 ⊢ {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
4		residm 6003	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑋) ↾ 𝑋) = (𝐹 ↾ 𝑋)
5	4	cnveqi 5854	. . . . . . . . 9 ⊢ ◡((𝐹 ↾ 𝑋) ↾ 𝑋) = ◡(𝐹 ↾ 𝑋)
6	5	imaeq1i 6053	. . . . . . . 8 ⊢ (◡((𝐹 ↾ 𝑋) ↾ 𝑋) “ 𝑠) = (◡(𝐹 ↾ 𝑋) “ 𝑠)
7		cnvresima 6226	. . . . . . . 8 ⊢ (◡((𝐹 ↾ 𝑋) ↾ 𝑋) “ 𝑠) = ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋)
8		cnvresima 6226	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝑋) “ 𝑠) = ((◡𝐹 “ 𝑠) ∩ 𝑋)
9	6, 7, 8	3eqtr3i 2791	. . . . . . 7 ⊢ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) = ((◡𝐹 “ 𝑠) ∩ 𝑋)
10	9	eleq1i 2851	. . . . . 6 ⊢ (((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽)
11	10	rabbii 3417	. . . . 5 ⊢ {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽}
12	3, 11	eqtr2i 2784	. . . 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 23922	. . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
15		resexg 6020	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝑋) ∈ V)
16	13	qtopval 23922	. . . . 5 ⊢ ((𝐽 ∈ V ∧ (𝐹 ↾ 𝑋) ∈ V) → (𝐽 qTop (𝐹 ↾ 𝑋)) = {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
17	15, 16	sylan2 605	. . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop (𝐹 ↾ 𝑋)) = {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
18	12, 14, 17	3eqtr4a 2821	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))
19	18	expcom 419	. 2 ⊢ (𝐹 ∈ 𝑉 → (𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋))))
20		df-qtop 17594	. . . . 5 ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
21	20	reldmmpo 7548	. . . 4 ⊢ Rel dom qTop
22	21	ovprc1 7453	. . 3 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop 𝐹) = ∅)
23	21	ovprc1 7453	. . 3 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop (𝐹 ↾ 𝑋)) = ∅)
24	22, 23	eqtr4d 2798	. 2 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))
25	19, 24	pm2.61d1 182	1 ⊢ (𝐹 ∈ 𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412  Vcvv 3450   ∩ cin 3898  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867  ^◡ccnv 5654   ↾ cres 5657   “ cima 5658  (class class class)co 7414   qTop cqtop 17590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-qtop 17594

This theorem is used by:  qtoptop2  23926