Theorem kqdisj 24060

Description: A version of imain 6630 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 6238	. . . . 5 ⊢ (𝐹 “ dom (𝐹 ↾ (𝐴 ∖ 𝑈))) = (𝐹 “ (𝐴 ∖ 𝑈))
2		dmres 6005	. . . . . . 7 ⊢ dom (𝐹 ↾ (𝐴 ∖ 𝑈)) = ((𝐴 ∖ 𝑈) ∩ dom 𝐹)
3		kqval.2	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
4	3	kqffn 24053	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
5	4	adantr 486	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝐹 Fn 𝑋)
6	5	fndmd 6649	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom 𝐹 = 𝑋)
7	6	ineq2d 4116	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐴 ∖ 𝑈) ∩ dom 𝐹) = ((𝐴 ∖ 𝑈) ∩ 𝑋))
8	2, 7	eqtrid 2807	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom (𝐹 ↾ (𝐴 ∖ 𝑈)) = ((𝐴 ∖ 𝑈) ∩ 𝑋))
9	8	imaeq2d 6054	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ dom (𝐹 ↾ (𝐴 ∖ 𝑈))) = (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)))
10	1, 9	eqtr3id 2809	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ (𝐴 ∖ 𝑈)) = (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)))
11		indif1 4226	. . . . . 6 ⊢ ((𝐴 ∖ 𝑈) ∩ 𝑋) = ((𝐴 ∩ 𝑋) ∖ 𝑈)
12		inss2 4133	. . . . . . 7 ⊢ (𝐴 ∩ 𝑋) ⊆ 𝑋
13		ssdif 4079	. . . . . . 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 3955	. . . . 5 ⊢ ((𝐴 ∖ 𝑈) ∩ 𝑋) ⊆ (𝑋 ∖ 𝑈)
16		imass2 6057	. . . . 5 ⊢ (((𝐴 ∖ 𝑈) ∩ 𝑋) ⊆ (𝑋 ∖ 𝑈) → (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)))
17	15, 16	mp1i 14	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)))
18	10, 17	eqsstrd 3943	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ (𝐴 ∖ 𝑈)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)))
19		sslin 4138	. . 3 ⊢ ((𝐹 “ (𝐴 ∖ 𝑈)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))))
20	18, 19	syl 18	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))))
21		eldifn 4067	. . . . . . 7 ⊢ (𝑤 ∈ (𝑋 ∖ 𝑈) → ¬ 𝑤 ∈ 𝑈)
22	21	adantl 487	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → ¬ 𝑤 ∈ 𝑈)
23		simpll 779	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝐽 ∈ (TopOn‘𝑋))
24		simplr 781	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝑈 ∈ 𝐽)
25		eldifi 4066	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑋 ∖ 𝑈) → 𝑤 ∈ 𝑋)
26	25	adantl 487	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝑤 ∈ 𝑋)
27	3	kqfvima 24058	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑈 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
28	23, 24, 26, 27	syl3anc 1398	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → (𝑤 ∈ 𝑈 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
29	22, 28	mtbid 327	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))
30	29	ralrimiva 3154	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))
31		difss 4071	. . . . 5 ⊢ (𝑋 ∖ 𝑈) ⊆ 𝑋
32		eleq1 2848	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑤) → (𝑧 ∈ (𝐹 “ 𝑈) ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
33	32	notbid 321	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑤) → (¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
34	33	ralima 7247	. . . . 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 598	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)))
36	30, 35	mpbird 260	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈))
37		disjr 4403	. . 3 ⊢ (((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅ ↔ ∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈))
38	36, 37	sylibr 237	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅)
39		sseq0 4353	. 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 596	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3896   ∖ cdif 4044   ∩ cin 4088  ∅c0 4277   ↦ cmpt 5185  dom cdm 5651   ↾ cres 5653   “ cima 5654   Fn wfn 6538  ‘cfv 6543  TopOnctopon 23237

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-topon 23238

This theorem is used by:  kqcldsat  24061  regr1lem  24067