Theorem kqdisj 21345

Description: A version of imain 5888 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
kqdisj	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) = ∅)

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.

Step	Hyp	Ref	Expression
1		imadmres 5544	. . . . 5 ⊢ (𝐹 “ dom (𝐹 ↾ (𝐴 ∖ 𝑈))) = (𝐹 “ (𝐴 ∖ 𝑈))
2		dmres 5339	. . . . . . 7 ⊢ dom (𝐹 ↾ (𝐴 ∖ 𝑈)) = ((𝐴 ∖ 𝑈) ∩ dom 𝐹)
3		kqval.2	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
4	3	kqffn 21338	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝐹 Fn 𝑋)
6		fndm 5904	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
7	5, 6	syl 17	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom 𝐹 = 𝑋)
8	7	ineq2d 3776	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐴 ∖ 𝑈) ∩ dom 𝐹) = ((𝐴 ∖ 𝑈) ∩ 𝑋))
9	2, 8	syl5eq 2656	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom (𝐹 ↾ (𝐴 ∖ 𝑈)) = ((𝐴 ∖ 𝑈) ∩ 𝑋))
10	9	imaeq2d 5385	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ dom (𝐹 ↾ (𝐴 ∖ 𝑈))) = (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)))
11	1, 10	syl5eqr 2658	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ (𝐴 ∖ 𝑈)) = (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)))
12		indif1 3830	. . . . . 6 ⊢ ((𝐴 ∖ 𝑈) ∩ 𝑋) = ((𝐴 ∩ 𝑋) ∖ 𝑈)
13		inss2 3796	. . . . . . 7 ⊢ (𝐴 ∩ 𝑋) ⊆ 𝑋
14		ssdif 3707	. . . . . . 7 ⊢ ((𝐴 ∩ 𝑋) ⊆ 𝑋 → ((𝐴 ∩ 𝑋) ∖ 𝑈) ⊆ (𝑋 ∖ 𝑈))
15	13, 14	ax-mp 5	. . . . . 6 ⊢ ((𝐴 ∩ 𝑋) ∖ 𝑈) ⊆ (𝑋 ∖ 𝑈)
16	12, 15	eqsstri 3598	. . . . 5 ⊢ ((𝐴 ∖ 𝑈) ∩ 𝑋) ⊆ (𝑋 ∖ 𝑈)
17		imass2 5420	. . . . 5 ⊢ (((𝐴 ∖ 𝑈) ∩ 𝑋) ⊆ (𝑋 ∖ 𝑈) → (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)))
18	16, 17	mp1i 13	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)))
19	11, 18	eqsstrd 3602	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ (𝐴 ∖ 𝑈)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)))
20		sslin 3801	. . 3 ⊢ ((𝐹 “ (𝐴 ∖ 𝑈)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))))
21	19, 20	syl 17	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))))
22		eldifn 3695	. . . . . . 7 ⊢ (𝑤 ∈ (𝑋 ∖ 𝑈) → ¬ 𝑤 ∈ 𝑈)
23	22	adantl 481	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → ¬ 𝑤 ∈ 𝑈)
24		simpll 786	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝐽 ∈ (TopOn‘𝑋))
25		simplr 788	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝑈 ∈ 𝐽)
26		eldifi 3694	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑋 ∖ 𝑈) → 𝑤 ∈ 𝑋)
27	26	adantl 481	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝑤 ∈ 𝑋)
28	3	kqfvima 21343	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑈 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
29	24, 25, 27, 28	syl3anc 1318	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → (𝑤 ∈ 𝑈 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
30	23, 29	mtbid 313	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))
31	30	ralrimiva 2949	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))
32		difss 3699	. . . . 5 ⊢ (𝑋 ∖ 𝑈) ⊆ 𝑋
33		eleq1 2676	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑤) → (𝑧 ∈ (𝐹 “ 𝑈) ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
34	33	notbid 307	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑤) → (¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
35	34	ralima 6402	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ (𝑋 ∖ 𝑈) ⊆ 𝑋) → (∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
36	5, 32, 35	sylancl 693	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
37	31, 36	mpbird 246	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈))
38		disjr 3970	. . 3 ⊢ (((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅ ↔ ∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈))
39	37, 38	sylibr 223	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅)
40		sseq0 3927	. 2 ⊢ ((((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ∧ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) = ∅)
41	21, 39, 40	syl2anc 691	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   ↦ cmpt 4643  dom cdm 5038   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ‘cfv 5804  TopOnctopon 20518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-topon 20523

This theorem is referenced by:  kqcldsat  21346  regr1lem  21352