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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 4328 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → ¬ (𝐽 ↾t 𝐴) = ∅) | |
2 | restfn 17379 | . . . . . . . . 9 ⊢ ↾t Fn (V × V) | |
3 | fndm 6646 | . . . . . . . . 9 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V)) | |
4 | 2, 3 | ax-mp 5 | . . . . . . . 8 ⊢ dom ↾t = (V × V) |
5 | 4 | ndmov 7588 | . . . . . . 7 ⊢ (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ∅) |
6 | 1, 5 | nsyl2 141 | . . . . . 6 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
7 | elrest 17382 | . . . . . 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 4224 | . . . . . 6 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝐴 | |
11 | sseq1 4002 | . . . . . 6 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∩ 𝐴) ⊆ 𝐴)) | |
12 | 10, 11 | mpbiri 258 | . . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴) |
13 | 12 | rexlimivw 3145 | . . . 4 ⊢ (∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴) |
14 | 9, 13 | syl 17 | . . 3 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ⊆ 𝐴) |
15 | velpw 4602 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
16 | 14, 15 | sylibr 233 | . 2 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ 𝒫 𝐴) |
17 | 16 | ssriv 3981 | 1 ⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 Vcvv 3468 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 𝒫 cpw 4597 × cxp 5667 dom cdm 5669 Fn wfn 6532 (class class class)co 7405 ↾t crest 17375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-rest 17377 |
This theorem is referenced by: 1stckgenlem 23412 prdstopn 23487 trfbas2 23702 trfil1 23745 trfil2 23746 fgtr 23749 trust 24089 zdis 24687 cnambfre 37049 dvdmsscn 45224 dvnmptconst 45229 dvnxpaek 45230 dvnmul 45231 dvnprodlem3 45236 |
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