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Theorem kqval 23734
Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqval (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 topontop 22919 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 id 22 . . . . 5 (𝑗 = 𝐽𝑗 = 𝐽)
3 unieq 4918 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
4 rabeq 3451 . . . . . 6 (𝑗 = 𝐽 → {𝑦𝑗𝑥𝑦} = {𝑦𝐽𝑥𝑦})
53, 4mpteq12dv 5233 . . . . 5 (𝑗 = 𝐽 → (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
62, 5oveq12d 7449 . . . 4 (𝑗 = 𝐽 → (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
7 df-kq 23702 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
8 ovex 7464 . . . 4 (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})) ∈ V
96, 7, 8fvmpt 7016 . . 3 (𝐽 ∈ Top → (KQ‘𝐽) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
101, 9syl 17 . 2 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
11 kqval.2 . . . 4 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
12 toponuni 22920 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1312mpteq1d 5237 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
1411, 13eqtrid 2789 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
1514oveq2d 7447 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
1610, 15eqtr4d 2780 1 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {crab 3436   cuni 4907  cmpt 5225  cfv 6561  (class class class)co 7431   qTop cqtop 17548  Topctop 22899  TopOnctopon 22916  KQckq 23701
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-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  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-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-topon 22917  df-kq 23702
This theorem is referenced by:  kqtopon  23735  kqid  23736  kqopn  23742  kqcld  23743  t0kq  23826
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