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Theorem qtopres 21311
 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 5351 . . . . . . 7 ((𝐹𝑋) “ 𝑋) = (𝐹𝑋)
21pweqi 4112 . . . . . 6 𝒫 ((𝐹𝑋) “ 𝑋) = 𝒫 (𝐹𝑋)
3 rabeq 3166 . . . . . 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 5350 . . . . . . . . . . 11 ((𝐹𝑋) ↾ 𝑋) = (𝐹𝑋)
65cnveqi 5219 . . . . . . . . . 10 ((𝐹𝑋) ↾ 𝑋) = (𝐹𝑋)
76imaeq1i 5382 . . . . . . . . 9 (((𝐹𝑋) ↾ 𝑋) “ 𝑠) = ((𝐹𝑋) “ 𝑠)
8 cnvresima 5541 . . . . . . . . 9 (((𝐹𝑋) ↾ 𝑋) “ 𝑠) = (((𝐹𝑋) “ 𝑠) ∩ 𝑋)
9 cnvresima 5541 . . . . . . . . 9 ((𝐹𝑋) “ 𝑠) = ((𝐹𝑠) ∩ 𝑋)
107, 8, 93eqtr3i 2640 . . . . . . . 8 (((𝐹𝑋) “ 𝑠) ∩ 𝑋) = ((𝐹𝑠) ∩ 𝑋)
1110eleq1i 2679 . . . . . . 7 ((((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽)
1211a1i 11 . . . . . 6 (𝑠 ∈ 𝒫 (𝐹𝑋) → ((((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽))
1312rabbiia 3161 . . . . 5 {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽}
144, 13eqtr2i 2633 . . . 4 {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽}
15 qtopval.1 . . . . 5 𝑋 = 𝐽
1615qtopval 21308 . . . 4 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
17 resexg 5362 . . . . 5 (𝐹𝑉 → (𝐹𝑋) ∈ V)
1815qtopval 21308 . . . . 5 ((𝐽 ∈ V ∧ (𝐹𝑋) ∈ V) → (𝐽 qTop (𝐹𝑋)) = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
1917, 18sylan2 490 . . . 4 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop (𝐹𝑋)) = {𝑠 ∈ 𝒫 ((𝐹𝑋) “ 𝑋) ∣ (((𝐹𝑋) “ 𝑠) ∩ 𝑋) ∈ 𝐽})
2014, 16, 193eqtr4a 2670 . . 3 ((𝐽 ∈ V ∧ 𝐹𝑉) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
2120expcom 450 . 2 (𝐹𝑉 → (𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋))))
22 df-qtop 15990 . . . . 5 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
2322reldmmpt2 6669 . . . 4 Rel dom qTop
2423ovprc1 6582 . . 3 𝐽 ∈ V → (𝐽 qTop 𝐹) = ∅)
2523ovprc1 6582 . . 3 𝐽 ∈ V → (𝐽 qTop (𝐹𝑋)) = ∅)
2624, 25eqtr4d 2647 . 2 𝐽 ∈ V → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
2721, 26pm2.61d1 170 1 (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∩ cin 3539  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ◡ccnv 5037   ↾ cres 5040   “ cima 5041  (class class class)co 6549   qTop cqtop 15986 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-qtop 15990 This theorem is referenced by:  qtoptop2  21312
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