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Theorem qtopval 21546
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 3243 . 2 (𝐽𝑉𝐽 ∈ V)
2 elex 3243 . 2 (𝐹𝑊𝐹 ∈ V)
3 imaexg 7145 . . . . 5 (𝐹 ∈ V → (𝐹𝑋) ∈ V)
4 pwexg 4880 . . . . 5 ((𝐹𝑋) ∈ V → 𝒫 (𝐹𝑋) ∈ V)
5 rabexg 4844 . . . . 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 4478 . . . . . . . 8 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
11 qtopval.1 . . . . . . . 8 𝑋 = 𝐽
1210, 11syl6eqr 2703 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝑋)
138, 12imaeq12d 5502 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓 𝑗) = (𝐹𝑋))
1413pweqd 4196 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝒫 (𝑓 𝑗) = 𝒫 (𝐹𝑋))
158cnveqd 5330 . . . . . . . 8 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
1615imaeq1d 5500 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓𝑠) = (𝐹𝑠))
1716, 12ineq12d 3848 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → ((𝑓𝑠) ∩ 𝑗) = ((𝐹𝑠) ∩ 𝑋))
1817, 9eleq12d 2724 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (((𝑓𝑠) ∩ 𝑗) ∈ 𝑗 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽))
1914, 18rabeqbidv 3226 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
20 df-qtop 16214 . . . 4 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
2119, 20ovmpt2ga 6832 . . 3 ((𝐽 ∈ V ∧ 𝐹 ∈ V ∧ {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
227, 21mpd3an3 1465 . 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 1523  wcel 2030  {crab 2945  Vcvv 3231  cin 3606  𝒫 cpw 4191   cuni 4468  ccnv 5142  cima 5146  (class class class)co 6690   qTop cqtop 16210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-qtop 16214
This theorem is referenced by:  qtopval2  21547  qtopres  21549  imastopn  21571
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