Theorem qtopres 24026

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 6058	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑋) “ 𝑋) = (𝐹 “ 𝑋)
2	1	pweqi 4572	. . . . . 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 5853	. . . . . . . . 9 ⊢ ◡((𝐹 ↾ 𝑋) ↾ 𝑋) = ◡(𝐹 ↾ 𝑋)
6	5	imaeq1i 6051	. . . . . . . 8 ⊢ (◡((𝐹 ↾ 𝑋) ↾ 𝑋) “ 𝑠) = (◡(𝐹 ↾ 𝑋) “ 𝑠)
7		cnvresima 6234	. . . . . . . 8 ⊢ (◡((𝐹 ↾ 𝑋) ↾ 𝑋) “ 𝑠) = ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋)
8		cnvresima 6234	. . . . . . . 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 24023	. . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
15		resexg 6018	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝑋) ∈ V)
16	13	qtopval 24023	. . . . 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 17688	. . . . 5 ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
21	20	reldmmpo 7558	. . . 4 ⊢ Rel dom qTop
22	21	ovprc1 7463	. . 3 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop 𝐹) = ∅)
23	21	ovprc1 7463	. . 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 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450   ∩ cin 4088  ∅c0 4277  𝒫 cpw 4556  ∪ cuni 4866  ^◡ccnv 5650   ↾ cres 5653   “ cima 5654  (class class class)co 7424   qTop cqtop 17684

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-qtop 17688

This theorem is used by:  qtoptop2  24027