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Theorem kqopn 23458
Description: The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
kqopn ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (𝐹 β€œ π‘ˆ) ∈ (KQβ€˜π½))
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑋,𝑦
Allowed substitution hints:   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)
Proof of Theorem kqopn
StepHypRef Expression
1 imassrn 6070 . . . 4 (𝐹 β€œ π‘ˆ) βŠ† ran 𝐹
21a1i 11 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (𝐹 β€œ π‘ˆ) βŠ† ran 𝐹)
3 kqval.2 . . . . 5 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
43kqsat 23455 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) = π‘ˆ)
5 simpr 485 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ π‘ˆ ∈ 𝐽)
64, 5eqeltrd 2833 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ∈ 𝐽)
73kqffn 23449 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
8 dffn4 6811 . . . . . 6 (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–ontoβ†’ran 𝐹)
97, 8sylib 217 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
109adantr 481 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
11 elqtop3 23427 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ ((𝐹 β€œ π‘ˆ) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 β€œ π‘ˆ) βŠ† ran 𝐹 ∧ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ∈ 𝐽)))
1210, 11syldan 591 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ ((𝐹 β€œ π‘ˆ) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 β€œ π‘ˆ) βŠ† ran 𝐹 ∧ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ∈ 𝐽)))
132, 6, 12mpbir2and 711 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (𝐹 β€œ π‘ˆ) ∈ (𝐽 qTop 𝐹))
143kqval 23450 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) = (𝐽 qTop 𝐹))
1514adantr 481 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (KQβ€˜π½) = (𝐽 qTop 𝐹))
1613, 15eleqtrrd 2836 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (𝐹 β€œ π‘ˆ) ∈ (KQβ€˜π½))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   βŠ† wss 3948   ↦ cmpt 5231  β—‘ccnv 5675  ran crn 5677   β€œ cima 5679   Fn wfn 6538  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7411   qTop cqtop 17453  TopOnctopon 22632  KQckq 23417
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  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-iun 4999  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-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-qtop 17457  df-topon 22633  df-kq 23418
This theorem is referenced by:  kqt0lem  23460  isr0  23461  regr1lem  23463  kqreglem1  23465  kqreglem2  23466  kqnrmlem1  23467  kqnrmlem2  23468
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