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Theorem qtopres 22225
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 5886 . . . . . . 7 ((𝐹𝑋) “ 𝑋) = (𝐹𝑋)
21pweqi 4546 . . . . . 6 𝒫 ((𝐹𝑋) “ 𝑋) = 𝒫 (𝐹𝑋)
32rabeqi 3488 . . . . 5 {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
4 residm 5885 . . . . . . . . . 10 ((𝐹𝑋) ↾ 𝑋) = (𝐹𝑋)
54cnveqi 5744 . . . . . . . . 9 ((𝐹𝑋) ↾ 𝑋) = (𝐹𝑋)
65imaeq1i 5924 . . . . . . . 8 (((𝐹𝑋) ↾ 𝑋) “ 𝑠) = ((𝐹𝑋) “ 𝑠)
7 cnvresima 6085 . . . . . . . 8 (((𝐹𝑋) ↾ 𝑋) “ 𝑠) = (((𝐹𝑋) “ 𝑠) ∩ 𝑋)
8 cnvresima 6085 . . . . . . . 8 ((𝐹𝑋) “ 𝑠) = ((𝐹𝑠) ∩ 𝑋)
96, 7, 83eqtr3i 2857 . . . . . . 7 (((𝐹𝑋) “ 𝑠) ∩ 𝑋) = ((𝐹𝑠) ∩ 𝑋)
109eleq1i 2908 . . . . . 6 ((((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽)
1110rabbii 3479 . . . . 5 {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽}
123, 11eqtr2i 2850 . . . 4 {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
13 qtopval.1 . . . . 5 𝑋 = 𝐽
1413qtopval 22222 . . . 4 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
15 resexg 5897 . . . . 5 (𝐹𝑉 → (𝐹𝑋) ∈ V)
1613qtopval 22222 . . . . 5 ((𝐽 ∈ V ∧ (𝐹𝑋) ∈ V) → (𝐽 qTop (𝐹𝑋)) = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
1715, 16sylan2 592 . . . 4 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop (𝐹𝑋)) = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
1812, 14, 173eqtr4a 2887 . . 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 16770 . . . . 5 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
2120reldmmpo 7275 . . . 4 Rel dom qTop
2221ovprc1 7187 . . 3 𝐽 ∈ V → (𝐽 qTop 𝐹) = ∅)
2321ovprc1 7187 . . 3 𝐽 ∈ V → (𝐽 qTop (𝐹𝑋)) = ∅)
2422, 23eqtr4d 2864 . 2 𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
2519, 24pm2.61d1 181 1 (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2107  {crab 3147  Vcvv 3500  cin 3939  c0 4295  𝒫 cpw 4542   cuni 4837  ccnv 5553  cres 5556  cima 5557  (class class class)co 7148   qTop cqtop 16766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-qtop 16770
This theorem is referenced by:  qtoptop2  22226
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