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| Mirrors > Home > MPE Home > Th. List > restopn2 | Structured version Visualization version GIF version | ||
| Description: If 𝐴 is open, then 𝐵 is open in 𝐴 iff it is an open subset of 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| Ref | Expression |
|---|---|
| restopn2 | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐵 ∈ (𝐽 ↾t 𝐴) ↔ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elssuni 4937 | . . . . 5 ⊢ (𝐵 ∈ (𝐽 ↾t 𝐴) → 𝐵 ⊆ ∪ (𝐽 ↾t 𝐴)) | |
| 2 | elssuni 4937 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
| 3 | eqid 2737 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 4 | 3 | restuni 23170 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
| 5 | 2, 4 | sylan2 593 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
| 6 | 5 | sseq2d 4016 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ ∪ (𝐽 ↾t 𝐴))) |
| 7 | 1, 6 | imbitrrid 246 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐵 ∈ (𝐽 ↾t 𝐴) → 𝐵 ⊆ 𝐴)) |
| 8 | 7 | pm4.71rd 562 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐵 ∈ (𝐽 ↾t 𝐴) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ (𝐽 ↾t 𝐴)))) |
| 9 | simpll 767 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ Top) | |
| 10 | simplr 769 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ 𝐽) | |
| 11 | ssidd 4007 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝐴) | |
| 12 | simpr 484 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴) | |
| 13 | restopnb 23183 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) ∧ (𝐴 ∈ 𝐽 ∧ 𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐴)) → (𝐵 ∈ 𝐽 ↔ 𝐵 ∈ (𝐽 ↾t 𝐴))) | |
| 14 | 9, 10, 10, 11, 12, 13 | syl23anc 1379 | . . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ 𝐽 ↔ 𝐵 ∈ (𝐽 ↾t 𝐴))) |
| 15 | 14 | pm5.32da 579 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → ((𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ 𝐽) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ (𝐽 ↾t 𝐴)))) |
| 16 | 8, 15 | bitr4d 282 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐵 ∈ (𝐽 ↾t 𝐴) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ 𝐽))) |
| 17 | 16 | biancomd 463 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐵 ∈ (𝐽 ↾t 𝐴) ↔ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ cuni 4907 (class class class)co 7431 ↾t crest 17465 Topctop 22899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-en 8986 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 |
| This theorem is referenced by: restdis 23186 perfopn 23193 llyrest 23493 nllyrest 23494 llyidm 23496 nllyidm 23497 lly1stc 23504 qtoprest 23725 xrtgioo 24828 lhop 26055 efopnlem2 26699 cvmopnlem 35283 cvmlift2lem9a 35308 cvmlift2lem9 35316 cvmlift3lem6 35329 restopn3 45156 restopnssd 45157 iscnrm3rlem6 48842 |
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