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Theorem kqval 21439
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 20641 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 id 22 . . . . 5 (𝑗 = 𝐽𝑗 = 𝐽)
3 unieq 4410 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
4 rabeq 3179 . . . . . 6 (𝑗 = 𝐽 → {𝑦𝑗𝑥𝑦} = {𝑦𝐽𝑥𝑦})
53, 4mpteq12dv 4693 . . . . 5 (𝑗 = 𝐽 → (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
62, 5oveq12d 6622 . . . 4 (𝑗 = 𝐽 → (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
7 df-kq 21407 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
8 ovex 6632 . . . 4 (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})) ∈ V
96, 7, 8fvmpt 6239 . . 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 20642 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1312mpteq1d 4698 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
1411, 13syl5eq 2667 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
1514oveq2d 6620 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
1610, 15eqtr4d 2658 1 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {crab 2911   cuni 4402  cmpt 4673  cfv 5847  (class class class)co 6604   qTop cqtop 16084  Topctop 20617  TopOnctopon 20618  KQckq 21406
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-topon 20623  df-kq 21407
This theorem is referenced by:  kqtopon  21440  kqid  21441  kqopn  21447  kqcld  21448  t0kq  21531
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