Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqdisj Structured version   Visualization version   GIF version

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
 Copyright terms: Public domain W3C validator