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Theorem kqdisj 23756
Description: A version of imain 6653 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 6256 . . . . 5 (𝐹 “ dom (𝐹 ↾ (𝐴𝑈))) = (𝐹 “ (𝐴𝑈))
2 dmres 6032 . . . . . . 7 dom (𝐹 ↾ (𝐴𝑈)) = ((𝐴𝑈) ∩ dom 𝐹)
3 kqval.2 . . . . . . . . . . 11 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
43kqffn 23749 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
54adantr 480 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝐹 Fn 𝑋)
65fndmd 6674 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → dom 𝐹 = 𝑋)
76ineq2d 4228 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐴𝑈) ∩ dom 𝐹) = ((𝐴𝑈) ∩ 𝑋))
82, 7eqtrid 2787 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → dom (𝐹 ↾ (𝐴𝑈)) = ((𝐴𝑈) ∩ 𝑋))
98imaeq2d 6080 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ dom (𝐹 ↾ (𝐴𝑈))) = (𝐹 “ ((𝐴𝑈) ∩ 𝑋)))
101, 9eqtr3id 2789 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐴𝑈)) = (𝐹 “ ((𝐴𝑈) ∩ 𝑋)))
11 indif1 4288 . . . . . 6 ((𝐴𝑈) ∩ 𝑋) = ((𝐴𝑋) ∖ 𝑈)
12 inss2 4246 . . . . . . 7 (𝐴𝑋) ⊆ 𝑋
13 ssdif 4154 . . . . . . 7 ((𝐴𝑋) ⊆ 𝑋 → ((𝐴𝑋) ∖ 𝑈) ⊆ (𝑋𝑈))
1412, 13ax-mp 5 . . . . . 6 ((𝐴𝑋) ∖ 𝑈) ⊆ (𝑋𝑈)
1511, 14eqsstri 4030 . . . . 5 ((𝐴𝑈) ∩ 𝑋) ⊆ (𝑋𝑈)
16 imass2 6123 . . . . 5 (((𝐴𝑈) ∩ 𝑋) ⊆ (𝑋𝑈) → (𝐹 “ ((𝐴𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋𝑈)))
1715, 16mp1i 13 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ ((𝐴𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋𝑈)))
1810, 17eqsstrd 4034 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐴𝑈)) ⊆ (𝐹 “ (𝑋𝑈)))
19 sslin 4251 . . 3 ((𝐹 “ (𝐴𝑈)) ⊆ (𝐹 “ (𝑋𝑈)) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) ⊆ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))))
2018, 19syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) ⊆ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))))
21 eldifn 4142 . . . . . . 7 (𝑤 ∈ (𝑋𝑈) → ¬ 𝑤𝑈)
2221adantl 481 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → ¬ 𝑤𝑈)
23 simpll 767 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝐽 ∈ (TopOn‘𝑋))
24 simplr 769 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝑈𝐽)
25 eldifi 4141 . . . . . . . 8 (𝑤 ∈ (𝑋𝑈) → 𝑤𝑋)
2625adantl 481 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝑤𝑋)
273kqfvima 23754 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝑤𝑋) → (𝑤𝑈 ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
2823, 24, 26, 27syl3anc 1370 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → (𝑤𝑈 ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
2922, 28mtbid 324 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → ¬ (𝐹𝑤) ∈ (𝐹𝑈))
3029ralrimiva 3144 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈))
31 difss 4146 . . . . 5 (𝑋𝑈) ⊆ 𝑋
32 eleq1 2827 . . . . . . 7 (𝑧 = (𝐹𝑤) → (𝑧 ∈ (𝐹𝑈) ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
3332notbid 318 . . . . . 6 (𝑧 = (𝐹𝑤) → (¬ 𝑧 ∈ (𝐹𝑈) ↔ ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
3433ralima 7257 . . . . 5 ((𝐹 Fn 𝑋 ∧ (𝑋𝑈) ⊆ 𝑋) → (∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈) ↔ ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
355, 31, 34sylancl 586 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈) ↔ ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
3630, 35mpbird 257 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈))
37 disjr 4457 . . 3 (((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅ ↔ ∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈))
3836, 37sylibr 234 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅)
39 sseq0 4409 . 2 ((((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) ⊆ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ∧ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
4020, 38, 39syl2anc 584 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cdif 3960  cin 3962  wss 3963  c0 4339  cmpt 5231  dom cdm 5689  cres 5691  cima 5692   Fn wfn 6558  cfv 6563  TopOnctopon 22932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-topon 22933
This theorem is referenced by:  kqcldsat  23757  regr1lem  23763
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