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Theorem kqsat 22311
 Description: Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 22297). (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 22305 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3 elpreima 6800 . . . . . 6 (𝐹 Fn 𝑋 → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
42, 3syl 17 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
54adantr 483 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
61kqfvima 22310 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝑧𝑋) → (𝑧𝑈 ↔ (𝐹𝑧) ∈ (𝐹𝑈)))
763expa 1114 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑧𝑋) → (𝑧𝑈 ↔ (𝐹𝑧) ∈ (𝐹𝑈)))
87biimprd 250 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → 𝑧𝑈))
98expimpd 456 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈)) → 𝑧𝑈))
105, 9sylbid 242 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) → 𝑧𝑈))
1110ssrdv 3948 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) ⊆ 𝑈)
12 toponss 21507 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝑋)
13 fndm 6427 . . . . . . 7 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
142, 13syl 17 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
1514adantr 483 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → dom 𝐹 = 𝑋)
1612, 15sseqtrrd 3983 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈 ⊆ dom 𝐹)
17 sseqin2 4166 . . . 4 (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹𝑈) = 𝑈)
1816, 17sylib 220 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (dom 𝐹𝑈) = 𝑈)
19 dminss 5982 . . 3 (dom 𝐹𝑈) ⊆ (𝐹 “ (𝐹𝑈))
2018, 19eqsstrrdi 3997 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈 ⊆ (𝐹 “ (𝐹𝑈)))
2111, 20eqssd 3959 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) = 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3129   ∩ cin 3908   ⊆ wss 3909   ↦ cmpt 5118  ◡ccnv 5526  dom cdm 5527   “ cima 5530   Fn wfn 6322  ‘cfv 6327  TopOnctopon 21490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-fv 6335  df-topon 21491 This theorem is referenced by:  kqopn  22314  kqreglem2  22322  kqnrmlem2  22324
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