Theorem qtopres 23758

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 6001	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑋) “ 𝑋) = (𝐹 “ 𝑋)
2	1	pweqi 4571	. . . . . 6 ⊢ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) = 𝒫 (𝐹 “ 𝑋)
3	2	rabeqi 3427	. . . . 5 ⊢ {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
4		residm 5996	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑋) ↾ 𝑋) = (𝐹 ↾ 𝑋)
5	4	cnveqi 5846	. . . . . . . . 9 ⊢ ◡((𝐹 ↾ 𝑋) ↾ 𝑋) = ◡(𝐹 ↾ 𝑋)
6	5	imaeq1i 6046	. . . . . . . 8 ⊢ (◡((𝐹 ↾ 𝑋) ↾ 𝑋) “ 𝑠) = (◡(𝐹 ↾ 𝑋) “ 𝑠)
7		cnvresima 6217	. . . . . . . 8 ⊢ (◡((𝐹 ↾ 𝑋) ↾ 𝑋) “ 𝑠) = ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋)
8		cnvresima 6217	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝑋) “ 𝑠) = ((◡𝐹 “ 𝑠) ∩ 𝑋)
9	6, 7, 8	3eqtr3i 2793	. . . . . . 7 ⊢ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) = ((◡𝐹 “ 𝑠) ∩ 𝑋)
10	9	eleq1i 2853	. . . . . 6 ⊢ (((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽)
11	10	rabbii 3419	. . . . 5 ⊢ {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽}
12	3, 11	eqtr2i 2786	. . . 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 23755	. . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
15		resexg 6013	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝑋) ∈ V)
16	13	qtopval 23755	. . . . 5 ⊢ ((𝐽 ∈ V ∧ (𝐹 ↾ 𝑋) ∈ V) → (𝐽 qTop (𝐹 ↾ 𝑋)) = {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
17	15, 16	sylan2 602	. . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop (𝐹 ↾ 𝑋)) = {𝑠 ∈ 𝒫 ((𝐹 ↾ 𝑋) “ 𝑋) ∣ ((◡(𝐹 ↾ 𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
18	12, 14, 17	3eqtr4a 2823	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ 𝑉) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))
19	18	expcom 417	. 2 ⊢ (𝐹 ∈ 𝑉 → (𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋))))
20		df-qtop 17537	. . . . 5 ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
21	20	reldmmpo 7530	. . . 4 ⊢ Rel dom qTop
22	21	ovprc1 7435	. . 3 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop 𝐹) = ∅)
23	21	ovprc1 7435	. . 3 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop (𝐹 ↾ 𝑋)) = ∅)
24	22, 23	eqtr4d 2800	. 2 ⊢ (¬ 𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))
25	19, 24	pm2.61d1 181	1 ⊢ (𝐹 ∈ 𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {crab 3414  Vcvv 3454   ∩ cin 3903  ∅c0 4285  𝒫 cpw 4555  ∪ cuni 4865  ^◡ccnv 5646   ↾ cres 5649   “ cima 5650  (class class class)co 7396   qTop cqtop 17533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-qtop 17537

This theorem is referenced by:  qtoptop2  23759