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Theorem qtopres 23706
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 (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))

Proof of Theorem qtopres
Dummy variables 𝑠 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resima 6033 . . . . . . 7 ((𝐹𝑋) “ 𝑋) = (𝐹𝑋)
21pweqi 4616 . . . . . 6 𝒫 ((𝐹𝑋) “ 𝑋) = 𝒫 (𝐹𝑋)
32rabeqi 3450 . . . . 5 {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
4 residm 6028 . . . . . . . . . 10 ((𝐹𝑋) ↾ 𝑋) = (𝐹𝑋)
54cnveqi 5885 . . . . . . . . 9 ((𝐹𝑋) ↾ 𝑋) = (𝐹𝑋)
65imaeq1i 6075 . . . . . . . 8 (((𝐹𝑋) ↾ 𝑋) “ 𝑠) = ((𝐹𝑋) “ 𝑠)
7 cnvresima 6250 . . . . . . . 8 (((𝐹𝑋) ↾ 𝑋) “ 𝑠) = (((𝐹𝑋) “ 𝑠) ∩ 𝑋)
8 cnvresima 6250 . . . . . . . 8 ((𝐹𝑋) “ 𝑠) = ((𝐹𝑠) ∩ 𝑋)
96, 7, 83eqtr3i 2773 . . . . . . 7 (((𝐹𝑋) “ 𝑠) ∩ 𝑋) = ((𝐹𝑠) ∩ 𝑋)
109eleq1i 2832 . . . . . 6 ((((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽)
1110rabbii 3442 . . . . 5 {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽}
123, 11eqtr2i 2766 . . . 4 {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
13 qtopval.1 . . . . 5 𝑋 = 𝐽
1413qtopval 23703 . . . 4 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
15 resexg 6045 . . . . 5 (𝐹𝑉 → (𝐹𝑋) ∈ V)
1613qtopval 23703 . . . . 5 ((𝐽 ∈ V ∧ (𝐹𝑋) ∈ V) → (𝐽 qTop (𝐹𝑋)) = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
1715, 16sylan2 593 . . . 4 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop (𝐹𝑋)) = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
1812, 14, 173eqtr4a 2803 . . 3 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
1918expcom 413 . 2 (𝐹𝑉 → (𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋))))
20 df-qtop 17552 . . . . 5 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
2120reldmmpo 7567 . . . 4 Rel dom qTop
2221ovprc1 7470 . . 3 𝐽 ∈ V → (𝐽 qTop 𝐹) = ∅)
2321ovprc1 7470 . . 3 𝐽 ∈ V → (𝐽 qTop (𝐹𝑋)) = ∅)
2422, 23eqtr4d 2780 . 2 𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
2519, 24pm2.61d1 180 1 (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cin 3950  c0 4333  𝒫 cpw 4600   cuni 4907  ccnv 5684  cres 5687  cima 5688  (class class class)co 7431   qTop cqtop 17548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17552
This theorem is referenced by:  qtoptop2  23707
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