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Theorem kqval 23550
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 22735 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 id 22 . . . . 5 (𝑗 = 𝐽𝑗 = 𝐽)
3 unieq 4919 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
4 rabeq 3445 . . . . . 6 (𝑗 = 𝐽 → {𝑦𝑗𝑥𝑦} = {𝑦𝐽𝑥𝑦})
53, 4mpteq12dv 5239 . . . . 5 (𝑗 = 𝐽 → (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
62, 5oveq12d 7430 . . . 4 (𝑗 = 𝐽 → (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
7 df-kq 23518 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
8 ovex 7445 . . . 4 (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})) ∈ V
96, 7, 8fvmpt 6998 . . 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 22736 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1312mpteq1d 5243 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
1411, 13eqtrid 2783 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
1514oveq2d 7428 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
1610, 15eqtr4d 2774 1 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  {crab 3431   cuni 4908  cmpt 5231  cfv 6543  (class class class)co 7412   qTop cqtop 17456  Topctop 22715  TopOnctopon 22732  KQckq 23517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-topon 22733  df-kq 23518
This theorem is referenced by:  kqtopon  23551  kqid  23552  kqopn  23558  kqcld  23559  t0kq  23642
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