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Theorem qtopval 21403
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 3203 . 2 (𝐽𝑉𝐽 ∈ V)
2 elex 3203 . 2 (𝐹𝑊𝐹 ∈ V)
3 imaexg 7051 . . . . 5 (𝐹 ∈ V → (𝐹𝑋) ∈ V)
4 pwexg 4815 . . . . 5 ((𝐹𝑋) ∈ V → 𝒫 (𝐹𝑋) ∈ V)
5 rabexg 4777 . . . . 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 482 . . 3 ((𝐽 ∈ V ∧ 𝐹 ∈ V) → {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V)
8 simpr 477 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
9 simpl 473 . . . . . . . . 9 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
109unieqd 4417 . . . . . . . 8 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
11 qtopval.1 . . . . . . . 8 𝑋 = 𝐽
1210, 11syl6eqr 2678 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝑋)
138, 12imaeq12d 5430 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓 𝑗) = (𝐹𝑋))
1413pweqd 4140 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝒫 (𝑓 𝑗) = 𝒫 (𝐹𝑋))
158cnveqd 5263 . . . . . . . 8 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
1615imaeq1d 5428 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓𝑠) = (𝐹𝑠))
1716, 12ineq12d 3798 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → ((𝑓𝑠) ∩ 𝑗) = ((𝐹𝑠) ∩ 𝑋))
1817, 9eleq12d 2698 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (((𝑓𝑠) ∩ 𝑗) ∈ 𝑗 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽))
1914, 18rabeqbidv 3186 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
20 df-qtop 16083 . . . 4 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
2119, 20ovmpt2ga 6744 . . 3 ((𝐽 ∈ V ∧ 𝐹 ∈ V ∧ {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
227, 21mpd3an3 1422 . 2 ((𝐽 ∈ V ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
231, 2, 22syl2an 494 1 ((𝐽𝑉𝐹𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  {crab 2916  Vcvv 3191  cin 3559  𝒫 cpw 4135   cuni 4407  ccnv 5078  cima 5082  (class class class)co 6605   qTop cqtop 16079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-qtop 16083
This theorem is referenced by:  qtopval2  21404  qtopres  21406  imastopn  21428
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