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Theorem kqdisj 23680
Description: A version of imain 6639 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 6240 . . . . 5 (𝐹 “ dom (𝐹 ↾ (𝐴𝑈))) = (𝐹 “ (𝐴𝑈))
2 dmres 6017 . . . . . . 7 dom (𝐹 ↾ (𝐴𝑈)) = ((𝐴𝑈) ∩ dom 𝐹)
3 kqval.2 . . . . . . . . . . 11 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
43kqffn 23673 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
54adantr 479 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝐹 Fn 𝑋)
65fndmd 6660 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → dom 𝐹 = 𝑋)
76ineq2d 4210 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐴𝑈) ∩ dom 𝐹) = ((𝐴𝑈) ∩ 𝑋))
82, 7eqtrid 2777 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → dom (𝐹 ↾ (𝐴𝑈)) = ((𝐴𝑈) ∩ 𝑋))
98imaeq2d 6064 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ dom (𝐹 ↾ (𝐴𝑈))) = (𝐹 “ ((𝐴𝑈) ∩ 𝑋)))
101, 9eqtr3id 2779 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐴𝑈)) = (𝐹 “ ((𝐴𝑈) ∩ 𝑋)))
11 indif1 4270 . . . . . 6 ((𝐴𝑈) ∩ 𝑋) = ((𝐴𝑋) ∖ 𝑈)
12 inss2 4228 . . . . . . 7 (𝐴𝑋) ⊆ 𝑋
13 ssdif 4136 . . . . . . 7 ((𝐴𝑋) ⊆ 𝑋 → ((𝐴𝑋) ∖ 𝑈) ⊆ (𝑋𝑈))
1412, 13ax-mp 5 . . . . . 6 ((𝐴𝑋) ∖ 𝑈) ⊆ (𝑋𝑈)
1511, 14eqsstri 4011 . . . . 5 ((𝐴𝑈) ∩ 𝑋) ⊆ (𝑋𝑈)
16 imass2 6107 . . . . 5 (((𝐴𝑈) ∩ 𝑋) ⊆ (𝑋𝑈) → (𝐹 “ ((𝐴𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋𝑈)))
1715, 16mp1i 13 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ ((𝐴𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋𝑈)))
1810, 17eqsstrd 4015 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐴𝑈)) ⊆ (𝐹 “ (𝑋𝑈)))
19 sslin 4233 . . 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 4124 . . . . . . 7 (𝑤 ∈ (𝑋𝑈) → ¬ 𝑤𝑈)
2221adantl 480 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → ¬ 𝑤𝑈)
23 simpll 765 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝐽 ∈ (TopOn‘𝑋))
24 simplr 767 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝑈𝐽)
25 eldifi 4123 . . . . . . . 8 (𝑤 ∈ (𝑋𝑈) → 𝑤𝑋)
2625adantl 480 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → 𝑤𝑋)
273kqfvima 23678 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝑤𝑋) → (𝑤𝑈 ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
2823, 24, 26, 27syl3anc 1368 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → (𝑤𝑈 ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
2922, 28mtbid 323 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) ∧ 𝑤 ∈ (𝑋𝑈)) → ¬ (𝐹𝑤) ∈ (𝐹𝑈))
3029ralrimiva 3135 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈))
31 difss 4128 . . . . 5 (𝑋𝑈) ⊆ 𝑋
32 eleq1 2813 . . . . . . 7 (𝑧 = (𝐹𝑤) → (𝑧 ∈ (𝐹𝑈) ↔ (𝐹𝑤) ∈ (𝐹𝑈)))
3332notbid 317 . . . . . 6 (𝑧 = (𝐹𝑤) → (¬ 𝑧 ∈ (𝐹𝑈) ↔ ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
3433ralima 7250 . . . . 5 ((𝐹 Fn 𝑋 ∧ (𝑋𝑈) ⊆ 𝑋) → (∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈) ↔ ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
355, 31, 34sylancl 584 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈) ↔ ∀𝑤 ∈ (𝑋𝑈) ¬ (𝐹𝑤) ∈ (𝐹𝑈)))
3630, 35mpbird 256 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈))
37 disjr 4451 . . 3 (((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅ ↔ ∀𝑧 ∈ (𝐹 “ (𝑋𝑈)) ¬ 𝑧 ∈ (𝐹𝑈))
3836, 37sylibr 233 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅)
39 sseq0 4401 . 2 ((((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) ⊆ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ∧ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
4020, 38, 39syl2anc 582 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  {crab 3418  cdif 3941  cin 3943  wss 3944  c0 4322  cmpt 5232  dom cdm 5678  cres 5680  cima 5681   Fn wfn 6544  cfv 6549  TopOnctopon 22856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-topon 22857
This theorem is referenced by:  kqcldsat  23681  regr1lem  23687
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