Theorem ntrss 15033

Description: Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)

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		simp3 1026	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆)
2		sspwb 4334	. . . . 5 ⊢ (𝑇 ⊆ 𝑆 ↔ 𝒫 𝑇 ⊆ 𝒫 𝑆)
3		sslin 3449	. . . . 5 ⊢ (𝒫 𝑇 ⊆ 𝒫 𝑆 → (𝐽 ∩ 𝒫 𝑇) ⊆ (𝐽 ∩ 𝒫 𝑆))
4	2, 3	sylbi 121	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝐽 ∩ 𝒫 𝑇) ⊆ (𝐽 ∩ 𝒫 𝑆))
5	4	unissd 3940	. . 3 ⊢ (𝑇 ⊆ 𝑆 → ∪ (𝐽 ∩ 𝒫 𝑇) ⊆ ∪ (𝐽 ∩ 𝒫 𝑆))
6	1, 5	syl 14	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ∪ (𝐽 ∩ 𝒫 𝑇) ⊆ ∪ (𝐽 ∩ 𝒫 𝑆))
7		simp1 1024	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝐽 ∈ Top)
8		simp2 1025	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ 𝑋)
9	1, 8	sstrd 3250	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
10		clscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
11	10	ntrval 15024	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋) → ((int‘𝐽)‘𝑇) = ∪ (𝐽 ∩ 𝒫 𝑇))
12	7, 9, 11	syl2anc 411	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑇) = ∪ (𝐽 ∩ 𝒫 𝑇))
13	10	ntrval 15024	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
14	7, 8, 13	syl2anc 411	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
15	6, 12, 14	3sstr4d 3285	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205   ∩ cin 3212   ⊆ wss 3213  𝒫 cpw 3671  ∪ cuni 3916  ‘cfv 5354  Topctop 14911  intcnt 15007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-top 14912  df-ntr 15010

This theorem is referenced by:  ntrin  15038  ntrcls0  15045