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Theorem kqsat 23455
Description: Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23441). (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
kqsat ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) = π‘ˆ)
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑋,𝑦
Allowed substitution hints:   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)
Proof of Theorem kqsat
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . . 7 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
21kqffn 23449 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
3 elpreima 7059 . . . . . 6 (𝐹 Fn 𝑋 β†’ (𝑧 ∈ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ↔ (𝑧 ∈ 𝑋 ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ))))
42, 3syl 17 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝑧 ∈ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ↔ (𝑧 ∈ 𝑋 ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ))))
54adantr 481 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (𝑧 ∈ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ↔ (𝑧 ∈ 𝑋 ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ))))
61kqfvima 23454 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) β†’ (𝑧 ∈ π‘ˆ ↔ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ)))
763expa 1118 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) ∧ 𝑧 ∈ 𝑋) β†’ (𝑧 ∈ π‘ˆ ↔ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ)))
87biimprd 247 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) ∧ 𝑧 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ) β†’ 𝑧 ∈ π‘ˆ))
98expimpd 454 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ ((𝑧 ∈ 𝑋 ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ)) β†’ 𝑧 ∈ π‘ˆ))
105, 9sylbid 239 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (𝑧 ∈ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) β†’ 𝑧 ∈ π‘ˆ))
1110ssrdv 3988 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) βŠ† π‘ˆ)
12 toponss 22649 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ π‘ˆ βŠ† 𝑋)
132fndmd 6654 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ dom 𝐹 = 𝑋)
1413adantr 481 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ dom 𝐹 = 𝑋)
1512, 14sseqtrrd 4023 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ π‘ˆ βŠ† dom 𝐹)
16 sseqin2 4215 . . . 4 (π‘ˆ βŠ† dom 𝐹 ↔ (dom 𝐹 ∩ π‘ˆ) = π‘ˆ)
1715, 16sylib 217 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (dom 𝐹 ∩ π‘ˆ) = π‘ˆ)
18 dminss 6152 . . 3 (dom 𝐹 ∩ π‘ˆ) βŠ† (◑𝐹 β€œ (𝐹 β€œ π‘ˆ))
1917, 18eqsstrrdi 4037 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ π‘ˆ βŠ† (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)))
2011, 19eqssd 3999 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) = π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   ∩ cin 3947   βŠ† wss 3948   ↦ cmpt 5231  β—‘ccnv 5675  dom cdm 5676   β€œ cima 5679   Fn wfn 6538  β€˜cfv 6543  TopOnctopon 22632
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-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-rab 3433  df-v 3476  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-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-topon 22633
This theorem is referenced by:  kqopn  23458  kqreglem2  23466  kqnrmlem2  23468
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