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Theorem qtopres 23201
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 6015 . . . . . . 7 ((𝐹𝑋) “ 𝑋) = (𝐹𝑋)
21pweqi 4618 . . . . . 6 𝒫 ((𝐹𝑋) “ 𝑋) = 𝒫 (𝐹𝑋)
32rabeqi 3445 . . . . 5 {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
4 residm 6014 . . . . . . . . . 10 ((𝐹𝑋) ↾ 𝑋) = (𝐹𝑋)
54cnveqi 5874 . . . . . . . . 9 ((𝐹𝑋) ↾ 𝑋) = (𝐹𝑋)
65imaeq1i 6056 . . . . . . . 8 (((𝐹𝑋) ↾ 𝑋) “ 𝑠) = ((𝐹𝑋) “ 𝑠)
7 cnvresima 6229 . . . . . . . 8 (((𝐹𝑋) ↾ 𝑋) “ 𝑠) = (((𝐹𝑋) “ 𝑠) ∩ 𝑋)
8 cnvresima 6229 . . . . . . . 8 ((𝐹𝑋) “ 𝑠) = ((𝐹𝑠) ∩ 𝑋)
96, 7, 83eqtr3i 2768 . . . . . . 7 (((𝐹𝑋) “ 𝑠) ∩ 𝑋) = ((𝐹𝑠) ∩ 𝑋)
109eleq1i 2824 . . . . . 6 ((((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽)
1110rabbii 3438 . . . . 5 {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽}
123, 11eqtr2i 2761 . . . 4 {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
13 qtopval.1 . . . . 5 𝑋 = 𝐽
1413qtopval 23198 . . . 4 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
15 resexg 6027 . . . . 5 (𝐹𝑉 → (𝐹𝑋) ∈ V)
1613qtopval 23198 . . . . 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 2798 . . 3 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
1918expcom 414 . 2 (𝐹𝑉 → (𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋))))
20 df-qtop 17452 . . . . 5 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
2120reldmmpo 7542 . . . 4 Rel dom qTop
2221ovprc1 7447 . . 3 𝐽 ∈ V → (𝐽 qTop 𝐹) = ∅)
2321ovprc1 7447 . . 3 𝐽 ∈ V → (𝐽 qTop (𝐹𝑋)) = ∅)
2422, 23eqtr4d 2775 . 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 396   = wceq 1541  wcel 2106  {crab 3432  Vcvv 3474  cin 3947  c0 4322  𝒫 cpw 4602   cuni 4908  ccnv 5675  cres 5678  cima 5679  (class class class)co 7408   qTop cqtop 17448
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-qtop 17452
This theorem is referenced by:  qtoptop2  23202
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