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Mirrors > Home > MPE Home > Th. List > 0rest | Structured version Visualization version GIF version |
Description: Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
0rest | ⊢ (∅ ↾t 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5300 | . . . 4 ⊢ ∅ ∈ V | |
2 | restval 17354 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴))) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴))) |
4 | mpt0 6679 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅ | |
5 | 4 | rneqi 5928 | . . . 4 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ran ∅ |
6 | rn0 5917 | . . . 4 ⊢ ran ∅ = ∅ | |
7 | 5, 6 | eqtri 2759 | . . 3 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅ |
8 | 3, 7 | eqtrdi 2787 | . 2 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅) |
9 | relxp 5687 | . . . 4 ⊢ Rel (V × V) | |
10 | restfn 17352 | . . . . . 6 ⊢ ↾t Fn (V × V) | |
11 | 10 | fndmi 6642 | . . . . 5 ⊢ dom ↾t = (V × V) |
12 | 11 | releqi 5769 | . . . 4 ⊢ (Rel dom ↾t ↔ Rel (V × V)) |
13 | 9, 12 | mpbir 230 | . . 3 ⊢ Rel dom ↾t |
14 | 13 | ovprc2 7433 | . 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅) |
15 | 8, 14 | pm2.61i 182 | 1 ⊢ (∅ ↾t 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∩ cin 3943 ∅c0 4318 ↦ cmpt 5224 × cxp 5667 dom cdm 5669 ran crn 5670 Rel wrel 5674 (class class class)co 7393 ↾t crest 17348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 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 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-rest 17350 |
This theorem is referenced by: firest 17360 topnval 17362 resstopn 22619 ussval 23693 bj-rest00 35766 |
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