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Theorem qtopval 23703
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 3501 . 2 (𝐽𝑉𝐽 ∈ V)
2 elex 3501 . 2 (𝐹𝑊𝐹 ∈ V)
3 imaexg 7935 . . . . 5 (𝐹 ∈ V → (𝐹𝑋) ∈ V)
4 pwexg 5378 . . . . 5 ((𝐹𝑋) ∈ V → 𝒫 (𝐹𝑋) ∈ V)
5 rabexg 5337 . . . . 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 484 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
9 simpl 482 . . . . . . . . 9 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
109unieqd 4920 . . . . . . . 8 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
11 qtopval.1 . . . . . . . 8 𝑋 = 𝐽
1210, 11eqtr4di 2795 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝑋)
138, 12imaeq12d 6079 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓 𝑗) = (𝐹𝑋))
1413pweqd 4617 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝒫 (𝑓 𝑗) = 𝒫 (𝐹𝑋))
158cnveqd 5886 . . . . . . . 8 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
1615imaeq1d 6077 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓𝑠) = (𝐹𝑠))
1716, 12ineq12d 4221 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → ((𝑓𝑠) ∩ 𝑗) = ((𝐹𝑠) ∩ 𝑋))
1817, 9eleq12d 2835 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (((𝑓𝑠) ∩ 𝑗) ∈ 𝑗 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽))
1914, 18rabeqbidv 3455 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
20 df-qtop 17552 . . . 4 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
2119, 20ovmpoga 7587 . . 3 ((𝐽 ∈ V ∧ 𝐹 ∈ V ∧ {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
227, 21mpd3an3 1464 . 2 ((𝐽 ∈ V ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
231, 2, 22syl2an 596 1 ((𝐽𝑉𝐹𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cin 3950  𝒫 cpw 4600   cuni 4907  ccnv 5684  cima 5688  (class class class)co 7431   qTop cqtop 17548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17552
This theorem is referenced by:  qtopval2  23704  qtopres  23706  imastopn  23728
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