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Theorem qtopres 21624
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 5541 . . . . . . 7 ((𝐹𝑋) “ 𝑋) = (𝐹𝑋)
21pweqi 4270 . . . . . 6 𝒫 ((𝐹𝑋) “ 𝑋) = 𝒫 (𝐹𝑋)
3 rabeq 3296 . . . . . 6 (𝒫 ((𝐹𝑋) “ 𝑋) = 𝒫 (𝐹𝑋) → {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
42, 3ax-mp 5 . . . . 5 {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
5 residm 5540 . . . . . . . . . 10 ((𝐹𝑋) ↾ 𝑋) = (𝐹𝑋)
65cnveqi 5404 . . . . . . . . 9 ((𝐹𝑋) ↾ 𝑋) = (𝐹𝑋)
76imaeq1i 5573 . . . . . . . 8 (((𝐹𝑋) ↾ 𝑋) “ 𝑠) = ((𝐹𝑋) “ 𝑠)
8 cnvresima 5736 . . . . . . . 8 (((𝐹𝑋) ↾ 𝑋) “ 𝑠) = (((𝐹𝑋) “ 𝑠) ∩ 𝑋)
9 cnvresima 5736 . . . . . . . 8 ((𝐹𝑋) “ 𝑠) = ((𝐹𝑠) ∩ 𝑋)
107, 8, 93eqtr3i 2754 . . . . . . 7 (((𝐹𝑋) “ 𝑠) ∩ 𝑋) = ((𝐹𝑠) ∩ 𝑋)
1110eleq1i 2794 . . . . . 6 ((((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽)
1211rabbii 3289 . . . . 5 {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽}
134, 12eqtr2i 2747 . . . 4 {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
14 qtopval.1 . . . . 5 𝑋 = 𝐽
1514qtopval 21621 . . . 4 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
16 resexg 5552 . . . . 5 (𝐹𝑉 → (𝐹𝑋) ∈ V)
1714qtopval 21621 . . . . 5 ((𝐽 ∈ V ∧ (𝐹𝑋) ∈ V) → (𝐽 qTop (𝐹𝑋)) = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
1816, 17sylan2 492 . . . 4 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop (𝐹𝑋)) = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
1913, 15, 183eqtr4a 2784 . . 3 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
2019expcom 450 . 2 (𝐹𝑉 → (𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋))))
21 df-qtop 16290 . . . . 5 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
2221reldmmpt2 6888 . . . 4 Rel dom qTop
2322ovprc1 6799 . . 3 𝐽 ∈ V → (𝐽 qTop 𝐹) = ∅)
2422ovprc1 6799 . . 3 𝐽 ∈ V → (𝐽 qTop (𝐹𝑋)) = ∅)
2523, 24eqtr4d 2761 . 2 𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
2620, 25pm2.61d1 171 1 (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1596  wcel 2103  {crab 3018  Vcvv 3304  cin 3679  c0 4023  𝒫 cpw 4266   cuni 4544  ccnv 5217  cres 5220  cima 5221  (class class class)co 6765   qTop cqtop 16286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-qtop 16290
This theorem is referenced by:  qtoptop2  21625
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