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Theorem kqval 22329
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 21516 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 id 22 . . . . 5 (𝑗 = 𝐽𝑗 = 𝐽)
3 unieq 4824 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
4 rabeq 3459 . . . . . 6 (𝑗 = 𝐽 → {𝑦𝑗𝑥𝑦} = {𝑦𝐽𝑥𝑦})
53, 4mpteq12dv 5127 . . . . 5 (𝑗 = 𝐽 → (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
62, 5oveq12d 7158 . . . 4 (𝑗 = 𝐽 → (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
7 df-kq 22297 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
8 ovex 7173 . . . 4 (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})) ∈ V
96, 7, 8fvmpt 6750 . . 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 21517 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1312mpteq1d 5131 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
1411, 13syl5eq 2869 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}))
1514oveq2d 7156 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
1610, 15eqtr4d 2860 1 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  {crab 3134   cuni 4813  cmpt 5122  cfv 6334  (class class class)co 7140   qTop cqtop 16767  Topctop 21496  TopOnctopon 21513  KQckq 22296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-topon 21514  df-kq 22297
This theorem is referenced by:  kqtopon  22330  kqid  22331  kqopn  22337  kqcld  22338  t0kq  22421
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