Theorem kqdisj 23740

Description: A version of imain 6651 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 6254	. . . . 5 ⊢ (𝐹 “ dom (𝐹 ↾ (𝐴 ∖ 𝑈))) = (𝐹 “ (𝐴 ∖ 𝑈))
2		dmres 6030	. . . . . . 7 ⊢ dom (𝐹 ↾ (𝐴 ∖ 𝑈)) = ((𝐴 ∖ 𝑈) ∩ dom 𝐹)
3		kqval.2	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
4	3	kqffn 23733	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝐹 Fn 𝑋)
6	5	fndmd 6673	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom 𝐹 = 𝑋)
7	6	ineq2d 4220	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐴 ∖ 𝑈) ∩ dom 𝐹) = ((𝐴 ∖ 𝑈) ∩ 𝑋))
8	2, 7	eqtrid 2789	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom (𝐹 ↾ (𝐴 ∖ 𝑈)) = ((𝐴 ∖ 𝑈) ∩ 𝑋))
9	8	imaeq2d 6078	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ dom (𝐹 ↾ (𝐴 ∖ 𝑈))) = (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)))
10	1, 9	eqtr3id 2791	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ (𝐴 ∖ 𝑈)) = (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)))
11		indif1 4282	. . . . . 6 ⊢ ((𝐴 ∖ 𝑈) ∩ 𝑋) = ((𝐴 ∩ 𝑋) ∖ 𝑈)
12		inss2 4238	. . . . . . 7 ⊢ (𝐴 ∩ 𝑋) ⊆ 𝑋
13		ssdif 4144	. . . . . . 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 4030	. . . . 5 ⊢ ((𝐴 ∖ 𝑈) ∩ 𝑋) ⊆ (𝑋 ∖ 𝑈)
16		imass2 6120	. . . . 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 4018	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ (𝐴 ∖ 𝑈)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)))
19		sslin 4243	. . 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 4132	. . . . . . 7 ⊢ (𝑤 ∈ (𝑋 ∖ 𝑈) → ¬ 𝑤 ∈ 𝑈)
22	21	adantl 481	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → ¬ 𝑤 ∈ 𝑈)
23		simpll 767	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝐽 ∈ (TopOn‘𝑋))
24		simplr 769	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝑈 ∈ 𝐽)
25		eldifi 4131	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑋 ∖ 𝑈) → 𝑤 ∈ 𝑋)
26	25	adantl 481	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝑤 ∈ 𝑋)
27	3	kqfvima 23738	. . . . . . 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 3146	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))
31		difss 4136	. . . . 5 ⊢ (𝑋 ∖ 𝑈) ⊆ 𝑋
32		eleq1 2829	. . . . . . 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 7257	. . . . 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 4451	. . 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 4403	. 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 2108  ∀wral 3061  {crab 3436   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333   ↦ cmpt 5225  dom cdm 5685   ↾ cres 5687   “ cima 5688   Fn wfn 6556  ‘cfv 6561  TopOnctopon 22916

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-topon 22917

This theorem is referenced by:  kqcldsat  23741  regr1lem  23747