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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 3879 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → ¬ (𝐽 ↾t 𝐴) = ∅) | |
2 | restfn 15908 | . . . . . . . . 9 ⊢ ↾t Fn (V × V) | |
3 | fndm 5904 | . . . . . . . . 9 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V)) | |
4 | 2, 3 | ax-mp 5 | . . . . . . . 8 ⊢ dom ↾t = (V × V) |
5 | 4 | ndmov 6716 | . . . . . . 7 ⊢ (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ∅) |
6 | 1, 5 | nsyl2 141 | . . . . . 6 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
7 | elrest 15911 | . . . . . 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 255 | . . . 4 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴)) |
10 | inss2 3796 | . . . . . 6 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝐴 | |
11 | sseq1 3589 | . . . . . 6 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∩ 𝐴) ⊆ 𝐴)) | |
12 | 10, 11 | mpbiri 247 | . . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴) |
13 | 12 | rexlimivw 3011 | . . . 4 ⊢ (∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴) |
14 | 9, 13 | syl 17 | . . 3 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ⊆ 𝐴) |
15 | selpw 4115 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
16 | 14, 15 | sylibr 223 | . 2 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ 𝒫 𝐴) |
17 | 16 | ssriv 3572 | 1 ⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 × cxp 5036 dom cdm 5038 Fn wfn 5799 (class class class)co 6549 ↾t crest 15904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-rest 15906 |
This theorem is referenced by: 1stckgenlem 21166 prdstopn 21241 trfbas2 21457 trfil1 21500 trfil2 21501 fgtr 21504 trust 21843 zdis 22427 cnambfre 32628 dvdmsscn 38826 dvnmptconst 38831 dvnxpaek 38832 dvnmul 38833 dvnprodlem3 38838 |
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