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Mirrors > Home > ILE Home > Th. List > ntrss | GIF version |
Description: Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
ntrss | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 999 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆) | |
2 | sspwb 4212 | . . . . 5 ⊢ (𝑇 ⊆ 𝑆 ↔ 𝒫 𝑇 ⊆ 𝒫 𝑆) | |
3 | sslin 3361 | . . . . 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 3831 | . . 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 997 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝐽 ∈ Top) | |
8 | simp2 998 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ 𝑋) | |
9 | 1, 8 | sstrd 3165 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋) |
10 | clscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
11 | 10 | ntrval 13243 | . . 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 13243 | . . 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 3200 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∩ cin 3128 ⊆ wss 3129 𝒫 cpw 3574 ∪ cuni 3807 ‘cfv 5211 Topctop 13128 intcnt 13226 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-top 13129 df-ntr 13229 |
This theorem is referenced by: ntrin 13257 ntrcls0 13264 |
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