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

Theorem kqdisj 21445
Description: A version of imain 5932 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 5586 . . . . 5 (𝐹 “ dom (𝐹 ↾ (𝐴𝑈))) = (𝐹 “ (𝐴𝑈))
2 dmres 5378 . . . . . . 7 dom (𝐹 ↾ (𝐴𝑈)) = ((𝐴𝑈) ∩ dom 𝐹)
3 kqval.2 . . . . . . . . . . 11 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
43kqffn 21438 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
54adantr 481 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝐹 Fn 𝑋)
6 fndm 5948 . . . . . . . . 9 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
75, 6syl 17 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → dom 𝐹 = 𝑋)
87ineq2d 3792 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐴𝑈) ∩ dom 𝐹) = ((𝐴𝑈) ∩ 𝑋))
92, 8syl5eq 2667 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → dom (𝐹 ↾ (𝐴𝑈)) = ((𝐴𝑈) ∩ 𝑋))
109imaeq2d 5425 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ dom (𝐹 ↾ (𝐴𝑈))) = (𝐹 “ ((𝐴𝑈) ∩ 𝑋)))
111, 10syl5eqr 2669 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐴𝑈)) = (𝐹 “ ((𝐴𝑈) ∩ 𝑋)))
12 indif1 3847 . . . . . 6 ((𝐴𝑈) ∩ 𝑋) = ((𝐴𝑋) ∖ 𝑈)
13 inss2 3812 . . . . . . 7 (𝐴𝑋) ⊆ 𝑋
14 ssdif 3723 . . . . . . 7 ((𝐴𝑋) ⊆ 𝑋 → ((𝐴𝑋) ∖ 𝑈) ⊆ (𝑋𝑈))
1513, 14ax-mp 5 . . . . . 6 ((𝐴𝑋) ∖ 𝑈) ⊆ (𝑋𝑈)
1612, 15eqsstri 3614 . . . . 5 ((𝐴𝑈) ∩ 𝑋) ⊆ (𝑋𝑈)
17 imass2 5460 . . . . 5 (((𝐴𝑈) ∩ 𝑋) ⊆ (𝑋𝑈) → (𝐹 “ ((𝐴𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋𝑈)))
1816, 17mp1i 13 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ ((𝐴𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋𝑈)))
1911, 18eqsstrd 3618 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐴𝑈)) ⊆ (𝐹 “ (𝑋𝑈)))
20 sslin 3817 . . 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 3711 . . . . . . 7 (𝑤 ∈ (𝑋𝑈) → ¬ 𝑤𝑈)
2322adantl 482 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → ¬ 𝑤𝑈)
24 simpll 789 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝐽 ∈ (TopOn‘𝑋))
25 simplr 791 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝑈𝐽)
26 eldifi 3710 . . . . . . . 8 (𝑤 ∈ (𝑋𝑈) → 𝑤𝑋)
2726adantl 482 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝑤𝑋)
283kqfvima 21443 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝑤𝑋) → (𝑤𝑈 ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
2924, 25, 27, 28syl3anc 1323 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → (𝑤𝑈 ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
3023, 29mtbid 314 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → ¬ (𝐹𝑤) ∈ (𝐹𝑈))
3130ralrimiva 2960 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈))
32 difss 3715 . . . . 5 (𝑋𝑈) ⊆ 𝑋
33 eleq1 2686 . . . . . . 7 (𝑧 = (𝐹𝑤) → (𝑧 ∈ (𝐹𝑈) ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
3433notbid 308 . . . . . 6 (𝑧 = (𝐹𝑤) → (¬ 𝑧 ∈ (𝐹𝑈) ↔ ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
3534ralima 6452 . . . . 5 ((𝐹 Fn 𝑋 ∧ (𝑋𝑈) ⊆ 𝑋) → (∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈) ↔ ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
365, 32, 35sylancl 693 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈) ↔ ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
3731, 36mpbird 247 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈))
38 disjr 3990 . . 3 (((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅ ↔ ∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈))
3937, 38sylibr 224 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅)
40 sseq0 3947 . 2 ((((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) ⊆ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ∧ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
4121, 39, 40syl2anc 692 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cdif 3552  cin 3554  wss 3555  c0 3891  cmpt 4673  dom cdm 5074  cres 5076  cima 5077   Fn wfn 5842  cfv 5847  TopOnctopon 20618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-topon 20623
This theorem is referenced by:  kqcldsat  21446  regr1lem  21452
  Copyright terms: Public domain W3C validator