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Theorem kqdisj 22332
Description: A version of imain 6432 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqdisj ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem kqdisj
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imadmres 6084 . . . . 5 (𝐹 “ dom (𝐹 ↾ (𝐴𝑈))) = (𝐹 “ (𝐴𝑈))
2 dmres 5868 . . . . . . 7 dom (𝐹 ↾ (𝐴𝑈)) = ((𝐴𝑈) ∩ dom 𝐹)
3 kqval.2 . . . . . . . . . . 11 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
43kqffn 22325 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
54adantr 483 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝐹 Fn 𝑋)
6 fndm 6448 . . . . . . . . 9 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
75, 6syl 17 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → dom 𝐹 = 𝑋)
87ineq2d 4187 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐴𝑈) ∩ dom 𝐹) = ((𝐴𝑈) ∩ 𝑋))
92, 8syl5eq 2866 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → dom (𝐹 ↾ (𝐴𝑈)) = ((𝐴𝑈) ∩ 𝑋))
109imaeq2d 5922 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ dom (𝐹 ↾ (𝐴𝑈))) = (𝐹 “ ((𝐴𝑈) ∩ 𝑋)))
111, 10syl5eqr 2868 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐴𝑈)) = (𝐹 “ ((𝐴𝑈) ∩ 𝑋)))
12 indif1 4246 . . . . . 6 ((𝐴𝑈) ∩ 𝑋) = ((𝐴𝑋) ∖ 𝑈)
13 inss2 4204 . . . . . . 7 (𝐴𝑋) ⊆ 𝑋
14 ssdif 4114 . . . . . . 7 ((𝐴𝑋) ⊆ 𝑋 → ((𝐴𝑋) ∖ 𝑈) ⊆ (𝑋𝑈))
1513, 14ax-mp 5 . . . . . 6 ((𝐴𝑋) ∖ 𝑈) ⊆ (𝑋𝑈)
1612, 15eqsstri 3999 . . . . 5 ((𝐴𝑈) ∩ 𝑋) ⊆ (𝑋𝑈)
17 imass2 5958 . . . . 5 (((𝐴𝑈) ∩ 𝑋) ⊆ (𝑋𝑈) → (𝐹 “ ((𝐴𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋𝑈)))
1816, 17mp1i 13 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ ((𝐴𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋𝑈)))
1911, 18eqsstrd 4003 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐴𝑈)) ⊆ (𝐹 “ (𝑋𝑈)))
20 sslin 4209 . . 3 ((𝐹 “ (𝐴𝑈)) ⊆ (𝐹 “ (𝑋𝑈)) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) ⊆ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))))
2119, 20syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) ⊆ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))))
22 eldifn 4102 . . . . . . 7 (𝑤 ∈ (𝑋𝑈) → ¬ 𝑤𝑈)
2322adantl 484 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → ¬ 𝑤𝑈)
24 simpll 765 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝐽 ∈ (TopOn‘𝑋))
25 simplr 767 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝑈𝐽)
26 eldifi 4101 . . . . . . . 8 (𝑤 ∈ (𝑋𝑈) → 𝑤𝑋)
2726adantl 484 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝑤𝑋)
283kqfvima 22330 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝑤𝑋) → (𝑤𝑈 ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
2924, 25, 27, 28syl3anc 1366 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → (𝑤𝑈 ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
3023, 29mtbid 326 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → ¬ (𝐹𝑤) ∈ (𝐹𝑈))
3130ralrimiva 3180 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈))
32 difss 4106 . . . . 5 (𝑋𝑈) ⊆ 𝑋
33 eleq1 2898 . . . . . . 7 (𝑧 = (𝐹𝑤) → (𝑧 ∈ (𝐹𝑈) ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
3433notbid 320 . . . . . 6 (𝑧 = (𝐹𝑤) → (¬ 𝑧 ∈ (𝐹𝑈) ↔ ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
3534ralima 6992 . . . . 5 ((𝐹 Fn 𝑋 ∧ (𝑋𝑈) ⊆ 𝑋) → (∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈) ↔ ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
365, 32, 35sylancl 588 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈) ↔ ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
3731, 36mpbird 259 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈))
38 disjr 4398 . . 3 (((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅ ↔ ∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈))
3937, 38sylibr 236 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅)
40 sseq0 4351 . 2 ((((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) ⊆ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ∧ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
4121, 39, 40syl2anc 586 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1531  wcel 2108  wral 3136  {crab 3140  cdif 3931  cin 3933  wss 3934  c0 4289  cmpt 5137  dom cdm 5548  cres 5550  cima 5551   Fn wfn 6343  cfv 6348  TopOnctopon 21510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-topon 21511
This theorem is referenced by:  kqcldsat  22333  regr1lem  22339
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