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| Mirrors > Home > MPE Home > Th. List > restsspw | Structured version Visualization version GIF version | ||
| Description: The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
| Ref | Expression |
|---|---|
| restsspw | ⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0i 4340 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → ¬ (𝐽 ↾t 𝐴) = ∅) | |
| 2 | restfn 17469 | . . . . . . . . 9 ⊢ ↾t Fn (V × V) | |
| 3 | fndm 6671 | . . . . . . . . 9 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V)) | |
| 4 | 2, 3 | ax-mp 5 | . . . . . . . 8 ⊢ dom ↾t = (V × V) |
| 5 | 4 | ndmov 7617 | . . . . . . 7 ⊢ (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ∅) |
| 6 | 1, 5 | nsyl2 141 | . . . . . 6 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
| 7 | elrest 17472 | . . . . . 6 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴))) | |
| 8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴))) |
| 9 | 8 | ibi 267 | . . . 4 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴)) |
| 10 | inss2 4238 | . . . . . 6 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝐴 | |
| 11 | sseq1 4009 | . . . . . 6 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∩ 𝐴) ⊆ 𝐴)) | |
| 12 | 10, 11 | mpbiri 258 | . . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴) |
| 13 | 12 | rexlimivw 3151 | . . . 4 ⊢ (∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴) |
| 14 | 9, 13 | syl 17 | . . 3 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ⊆ 𝐴) |
| 15 | velpw 4605 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
| 16 | 14, 15 | sylibr 234 | . 2 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ 𝒫 𝐴) |
| 17 | 16 | ssriv 3987 | 1 ⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 × cxp 5683 dom cdm 5685 Fn wfn 6556 (class class class)co 7431 ↾t crest 17465 |
| 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-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 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-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-1st 8014 df-2nd 8015 df-rest 17467 |
| This theorem is referenced by: 1stckgenlem 23561 prdstopn 23636 trfbas2 23851 trfil1 23894 trfil2 23895 fgtr 23898 trust 24238 zdis 24838 cnambfre 37675 dvdmsscn 45951 dvnmptconst 45956 dvnxpaek 45957 dvnmul 45958 dvnprodlem3 45963 |
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