Theorem kqdisj 23595

Description: A version of imain 6585 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 6195	. . . . 5 ⊢ (𝐹 “ dom (𝐹 ↾ (𝐴 ∖ 𝑈))) = (𝐹 “ (𝐴 ∖ 𝑈))
2		dmres 5972	. . . . . . 7 ⊢ dom (𝐹 ↾ (𝐴 ∖ 𝑈)) = ((𝐴 ∖ 𝑈) ∩ dom 𝐹)
3		kqval.2	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
4	3	kqffn 23588	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝐹 Fn 𝑋)
6	5	fndmd 6605	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom 𝐹 = 𝑋)
7	6	ineq2d 4179	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐴 ∖ 𝑈) ∩ dom 𝐹) = ((𝐴 ∖ 𝑈) ∩ 𝑋))
8	2, 7	eqtrid 2776	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom (𝐹 ↾ (𝐴 ∖ 𝑈)) = ((𝐴 ∖ 𝑈) ∩ 𝑋))
9	8	imaeq2d 6020	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ dom (𝐹 ↾ (𝐴 ∖ 𝑈))) = (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)))
10	1, 9	eqtr3id 2778	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ (𝐴 ∖ 𝑈)) = (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)))
11		indif1 4241	. . . . . 6 ⊢ ((𝐴 ∖ 𝑈) ∩ 𝑋) = ((𝐴 ∩ 𝑋) ∖ 𝑈)
12		inss2 4197	. . . . . . 7 ⊢ (𝐴 ∩ 𝑋) ⊆ 𝑋
13		ssdif 4103	. . . . . . 7 ⊢ ((𝐴 ∩ 𝑋) ⊆ 𝑋 → ((𝐴 ∩ 𝑋) ∖ 𝑈) ⊆ (𝑋 ∖ 𝑈))
14	12, 13	ax-mp 5	. . . . . 6 ⊢ ((𝐴 ∩ 𝑋) ∖ 𝑈) ⊆ (𝑋 ∖ 𝑈)
15	11, 14	eqsstri 3990	. . . . 5 ⊢ ((𝐴 ∖ 𝑈) ∩ 𝑋) ⊆ (𝑋 ∖ 𝑈)
16		imass2 6062	. . . . 5 ⊢ (((𝐴 ∖ 𝑈) ∩ 𝑋) ⊆ (𝑋 ∖ 𝑈) → (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)))
17	15, 16	mp1i 13	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)))
18	10, 17	eqsstrd 3978	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ (𝐴 ∖ 𝑈)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)))
19		sslin 4202	. . 3 ⊢ ((𝐹 “ (𝐴 ∖ 𝑈)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))))
20	18, 19	syl 17	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))))
21		eldifn 4091	. . . . . . 7 ⊢ (𝑤 ∈ (𝑋 ∖ 𝑈) → ¬ 𝑤 ∈ 𝑈)
22	21	adantl 481	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → ¬ 𝑤 ∈ 𝑈)
23		simpll 766	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝐽 ∈ (TopOn‘𝑋))
24		simplr 768	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝑈 ∈ 𝐽)
25		eldifi 4090	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑋 ∖ 𝑈) → 𝑤 ∈ 𝑋)
26	25	adantl 481	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝑤 ∈ 𝑋)
27	3	kqfvima 23593	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑈 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
28	23, 24, 26, 27	syl3anc 1373	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → (𝑤 ∈ 𝑈 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
29	22, 28	mtbid 324	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))
30	29	ralrimiva 3125	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))
31		difss 4095	. . . . 5 ⊢ (𝑋 ∖ 𝑈) ⊆ 𝑋
32		eleq1 2816	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑤) → (𝑧 ∈ (𝐹 “ 𝑈) ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
33	32	notbid 318	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑤) → (¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
34	33	ralima 7193	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ (𝑋 ∖ 𝑈) ⊆ 𝑋) → (∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
35	5, 31, 34	sylancl 586	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
36	30, 35	mpbird 257	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈))
37		disjr 4410	. . 3 ⊢ (((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅ ↔ ∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈))
38	36, 37	sylibr 234	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅)
39		sseq0 4362	. 2 ⊢ ((((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ∧ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) = ∅)
40	20, 38, 39	syl2anc 584	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292   ↦ cmpt 5183  dom cdm 5631   ↾ cres 5633   “ cima 5634   Fn wfn 6494  ‘cfv 6499  TopOnctopon 22773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-topon 22774

This theorem is referenced by:  kqcldsat  23596  regr1lem  23602