Theorem ntrss 23173

Description: Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ntrss	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))

Proof of Theorem ntrss

Step	Hyp	Ref	Expression
1		sscon 4099	. . . . . . 7 ⊢ (𝑇 ⊆ 𝑆 → (𝑋 ∖ 𝑆) ⊆ (𝑋 ∖ 𝑇))
2	1	adantl 486	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑋 ∖ 𝑆) ⊆ (𝑋 ∖ 𝑇))
3		difss 4092	. . . . . 6 ⊢ (𝑋 ∖ 𝑇) ⊆ 𝑋
4	2, 3	jctil 528	. . . . 5 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((𝑋 ∖ 𝑇) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ⊆ (𝑋 ∖ 𝑇)))
5		clscld.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
6	5	clsss 23172	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑇) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ⊆ (𝑋 ∖ 𝑇)) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) ⊆ ((cls‘𝐽)‘(𝑋 ∖ 𝑇)))
7	6	3expb 1136	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ ((𝑋 ∖ 𝑇) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ⊆ (𝑋 ∖ 𝑇))) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) ⊆ ((cls‘𝐽)‘(𝑋 ∖ 𝑇)))
8	4, 7	sylan2 604	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆)) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) ⊆ ((cls‘𝐽)‘(𝑋 ∖ 𝑇)))
9	8	sscond 4102	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑇))) ⊆ (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))))
10		sstr2 3946	. . . . 5 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑋 → 𝑇 ⊆ 𝑋))
11	10	impcom 412	. . . 4 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
12	5	ntrval2 23169	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋) → ((int‘𝐽)‘𝑇) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑇))))
13	11, 12	sylan2 604	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆)) → ((int‘𝐽)‘𝑇) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑇))))
14	5	ntrval2 23169	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))))
15	14	adantrr 729	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆)) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))))
16	9, 13, 15	3sstr4d 3994	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆)) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
17	16	3impb 1130	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ∖ cdif 3904   ⊆ wss 3907  ∪ cuni 4868  ‘cfv 6525  Topctop 23011  intcnt 23135  clsccl 23136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-top 23012  df-cld 23137  df-ntr 23138  df-cls 23139

This theorem is referenced by:  ntrin  23179  ntrcls0  23194  dvreslem  26029  dvres2lem  26030  dvaddbr  26058  dvmulbr  26059  dvcnvrelem2  26138  ntruni  36700  cldregopn  36704  limciccioolb  46195  limcicciooub  46209  cncfiooicclem1  46465