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Theorem kqval 23750
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 22935 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 id 22 . . . . 5 (𝑗 = 𝐽𝑗 = 𝐽)
3 unieq 4923 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
4 rabeq 3448 . . . . . 6 (𝑗 = 𝐽 → {𝑦𝑗𝑥𝑦} = {𝑦𝐽𝑥𝑦})
53, 4mpteq12dv 5239 . . . . 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 23718 . . . 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 22936 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1312mpteq1d 5243 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
1411, 13eqtrid 2787 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
1514oveq2d 7447 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
1610, 15eqtr4d 2778 1 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433   cuni 4912  cmpt 5231  cfv 6563  (class class class)co 7431   qTop cqtop 17550  Topctop 22915  TopOnctopon 22932  KQckq 23717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-topon 22933  df-kq 23718
This theorem is referenced by:  kqtopon  23751  kqid  23752  kqopn  23758  kqcld  23759  t0kq  23842
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