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Theorem qtopval 21308
 Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
qtopval.1 𝑋 = 𝐽
Assertion
Ref Expression
qtopval ((𝐽𝑉𝐹𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
Distinct variable groups:   𝐹,𝑠   𝐽,𝑠   𝑉,𝑠   𝑋,𝑠
Allowed substitution hint:   𝑊(𝑠)

Proof of Theorem qtopval
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐽𝑉𝐽 ∈ V)
2 elex 3185 . 2 (𝐹𝑊𝐹 ∈ V)
3 imaexg 6995 . . . . 5 (𝐹 ∈ V → (𝐹𝑋) ∈ V)
4 pwexg 4776 . . . . 5 ((𝐹𝑋) ∈ V → 𝒫 (𝐹𝑋) ∈ V)
5 rabexg 4739 . . . . 5 (𝒫 (𝐹𝑋) ∈ V → {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V)
63, 4, 53syl 18 . . . 4 (𝐹 ∈ V → {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V)
76adantl 481 . . 3 ((𝐽 ∈ V ∧ 𝐹 ∈ V) → {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V)
8 simpr 476 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
9 simpl 472 . . . . . . . . 9 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
109unieqd 4382 . . . . . . . 8 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
11 qtopval.1 . . . . . . . 8 𝑋 = 𝐽
1210, 11syl6eqr 2662 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝑋)
138, 12imaeq12d 5386 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓 𝑗) = (𝐹𝑋))
1413pweqd 4113 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝒫 (𝑓 𝑗) = 𝒫 (𝐹𝑋))
158cnveqd 5220 . . . . . . . 8 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
1615imaeq1d 5384 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓𝑠) = (𝐹𝑠))
1716, 12ineq12d 3777 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → ((𝑓𝑠) ∩ 𝑗) = ((𝐹𝑠) ∩ 𝑋))
1817, 9eleq12d 2682 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (((𝑓𝑠) ∩ 𝑗) ∈ 𝑗 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽))
1914, 18rabeqbidv 3168 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
20 df-qtop 15990 . . . 4 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
2119, 20ovmpt2ga 6688 . . 3 ((𝐽 ∈ V ∧ 𝐹 ∈ V ∧ {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
227, 21mpd3an3 1417 . 2 ((𝐽 ∈ V ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
231, 2, 22syl2an 493 1 ((𝐽𝑉𝐹𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∩ cin 3539  𝒫 cpw 4108  ∪ cuni 4372  ◡ccnv 5037   “ 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:  qtopval2  21309  qtopres  21311  imastopn  21333
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