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Theorem kqsat 21755
Description: Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 21741). (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqsat ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) = 𝑈)
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem kqsat
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . . 7 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqffn 21749 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3 elpreima 6482 . . . . . 6 (𝐹 Fn 𝑋 → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
42, 3syl 17 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
54adantr 466 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
61kqfvima 21754 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝑧𝑋) → (𝑧𝑈 ↔ (𝐹𝑧) ∈ (𝐹𝑈)))
763expa 1111 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑧𝑋) → (𝑧𝑈 ↔ (𝐹𝑧) ∈ (𝐹𝑈)))
87biimprd 238 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → 𝑧𝑈))
98expimpd 441 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈)) → 𝑧𝑈))
105, 9sylbid 230 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) → 𝑧𝑈))
1110ssrdv 3758 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) ⊆ 𝑈)
12 toponss 20952 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝑋)
13 fndm 6129 . . . . . . 7 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
142, 13syl 17 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
1514adantr 466 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → dom 𝐹 = 𝑋)
1612, 15sseqtr4d 3791 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈 ⊆ dom 𝐹)
17 sseqin2 3968 . . . 4 (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹𝑈) = 𝑈)
1816, 17sylib 208 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (dom 𝐹𝑈) = 𝑈)
19 dminss 5687 . . 3 (dom 𝐹𝑈) ⊆ (𝐹 “ (𝐹𝑈))
2018, 19syl6eqssr 3805 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈 ⊆ (𝐹 “ (𝐹𝑈)))
2111, 20eqssd 3769 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {crab 3065  cin 3722  wss 3723  cmpt 4864  ccnv 5249  dom cdm 5250  cima 5253   Fn wfn 6025  cfv 6030  TopOnctopon 20935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-topon 20936
This theorem is referenced by:  kqopn  21758  kqreglem2  21766  kqnrmlem2  21768
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