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Theorem qtopval 23069
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 3465 . 2 (𝐽𝑉𝐽 ∈ V)
2 elex 3465 . 2 (𝐹𝑊𝐹 ∈ V)
3 imaexg 7856 . . . . 5 (𝐹 ∈ V → (𝐹𝑋) ∈ V)
4 pwexg 5337 . . . . 5 ((𝐹𝑋) ∈ V → 𝒫 (𝐹𝑋) ∈ V)
5 rabexg 5292 . . . . 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 483 . . 3 ((𝐽 ∈ V ∧ 𝐹 ∈ V) → {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V)
8 simpr 486 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
9 simpl 484 . . . . . . . . 9 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
109unieqd 4883 . . . . . . . 8 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
11 qtopval.1 . . . . . . . 8 𝑋 = 𝐽
1210, 11eqtr4di 2791 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝑋)
138, 12imaeq12d 6018 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓 𝑗) = (𝐹𝑋))
1413pweqd 4581 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝒫 (𝑓 𝑗) = 𝒫 (𝐹𝑋))
158cnveqd 5835 . . . . . . . 8 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
1615imaeq1d 6016 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓𝑠) = (𝐹𝑠))
1716, 12ineq12d 4177 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → ((𝑓𝑠) ∩ 𝑗) = ((𝐹𝑠) ∩ 𝑋))
1817, 9eleq12d 2828 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (((𝑓𝑠) ∩ 𝑗) ∈ 𝑗 ↔ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽))
1914, 18rabeqbidv 3423 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗} = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
20 df-qtop 17397 . . . 4 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
2119, 20ovmpoga 7513 . . 3 ((𝐽 ∈ V ∧ 𝐹 ∈ V ∧ {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
227, 21mpd3an3 1463 . 2 ((𝐽 ∈ V ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
231, 2, 22syl2an 597 1 ((𝐽𝑉𝐹𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {crab 3406  Vcvv 3447  cin 3913  𝒫 cpw 4564   cuni 4869  ccnv 5636  cima 5640  (class class class)co 7361   qTop cqtop 17393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-qtop 17397
This theorem is referenced by:  qtopval2  23070  qtopres  23072  imastopn  23094
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