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Theorem ntrss 21655
Description: Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
ntrss ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))

Proof of Theorem ntrss
StepHypRef Expression
1 sscon 4113 . . . . . . 7 (𝑇𝑆 → (𝑋𝑆) ⊆ (𝑋𝑇))
21adantl 484 . . . . . 6 ((𝑆𝑋𝑇𝑆) → (𝑋𝑆) ⊆ (𝑋𝑇))
3 difss 4106 . . . . . 6 (𝑋𝑇) ⊆ 𝑋
42, 3jctil 522 . . . . 5 ((𝑆𝑋𝑇𝑆) → ((𝑋𝑇) ⊆ 𝑋 ∧ (𝑋𝑆) ⊆ (𝑋𝑇)))
5 clscld.1 . . . . . . 7 𝑋 = 𝐽
65clsss 21654 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑋𝑇) ⊆ 𝑋 ∧ (𝑋𝑆) ⊆ (𝑋𝑇)) → ((cls‘𝐽)‘(𝑋𝑆)) ⊆ ((cls‘𝐽)‘(𝑋𝑇)))
763expb 1115 . . . . 5 ((𝐽 ∈ Top ∧ ((𝑋𝑇) ⊆ 𝑋 ∧ (𝑋𝑆) ⊆ (𝑋𝑇))) → ((cls‘𝐽)‘(𝑋𝑆)) ⊆ ((cls‘𝐽)‘(𝑋𝑇)))
84, 7sylan2 594 . . . 4 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → ((cls‘𝐽)‘(𝑋𝑆)) ⊆ ((cls‘𝐽)‘(𝑋𝑇)))
98sscond 4116 . . 3 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑇))) ⊆ (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
10 sstr2 3972 . . . . 5 (𝑇𝑆 → (𝑆𝑋𝑇𝑋))
1110impcom 410 . . . 4 ((𝑆𝑋𝑇𝑆) → 𝑇𝑋)
125ntrval2 21651 . . . 4 ((𝐽 ∈ Top ∧ 𝑇𝑋) → ((int‘𝐽)‘𝑇) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑇))))
1311, 12sylan2 594 . . 3 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → ((int‘𝐽)‘𝑇) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑇))))
145ntrval2 21651 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
1514adantrr 715 . . 3 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
169, 13, 153sstr4d 4012 . 2 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
17163impb 1110 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1082   = wceq 1531  wcel 2108  cdif 3931  wss 3934   cuni 4830  cfv 6348  Topctop 21493  intcnt 21617  clsccl 21618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21494  df-cld 21619  df-ntr 21620  df-cls 21621
This theorem is referenced by:  ntrin  21661  ntrcls0  21676  dvreslem  24499  dvres2lem  24500  dvaddbr  24527  dvmulbr  24528  dvcnvrelem2  24607  ntruni  33668  cldregopn  33672  limciccioolb  41891  limcicciooub  41907  cncfiooicclem1  42165
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