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Theorem kqval 23100
Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
kqval (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) = (𝐽 qTop 𝐹))
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑋,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦)

Proof of Theorem kqval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 topontop 22285 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 id 22 . . . . 5 (𝑗 = 𝐽 β†’ 𝑗 = 𝐽)
3 unieq 4880 . . . . . 6 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
4 rabeq 3420 . . . . . 6 (𝑗 = 𝐽 β†’ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
53, 4mpteq12dv 5200 . . . . 5 (𝑗 = 𝐽 β†’ (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}))
62, 5oveq12d 7379 . . . 4 (𝑗 = 𝐽 β†’ (𝑗 qTop (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦})) = (𝐽 qTop (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})))
7 df-kq 23068 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦})))
8 ovex 7394 . . . 4 (𝐽 qTop (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})) ∈ V
96, 7, 8fvmpt 6952 . . 3 (𝐽 ∈ Top β†’ (KQβ€˜π½) = (𝐽 qTop (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})))
101, 9syl 17 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) = (𝐽 qTop (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})))
11 kqval.2 . . . 4 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
12 toponuni 22286 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1312mpteq1d 5204 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}))
1411, 13eqtrid 2785 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}))
1514oveq2d 7377 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 qTop 𝐹) = (𝐽 qTop (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})))
1610, 15eqtr4d 2776 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3406  βˆͺ cuni 4869   ↦ cmpt 5192  β€˜cfv 6500  (class class class)co 7361   qTop cqtop 17393  Topctop 22265  TopOnctopon 22282  KQckq 23067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-topon 22283  df-kq 23068
This theorem is referenced by:  kqtopon  23101  kqid  23102  kqopn  23108  kqcld  23109  t0kq  23192
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