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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 5307 | . . . 4 ⊢ ∅ ∈ V | |
2 | restval 17407 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴))) | |
3 | 1, 2 | mpan 689 | . . 3 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴))) |
4 | mpt0 6697 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅ | |
5 | 4 | rneqi 5939 | . . . 4 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ran ∅ |
6 | rn0 5928 | . . . 4 ⊢ ran ∅ = ∅ | |
7 | 5, 6 | eqtri 2756 | . . 3 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅ |
8 | 3, 7 | eqtrdi 2784 | . 2 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅) |
9 | relxp 5696 | . . . 4 ⊢ Rel (V × V) | |
10 | restfn 17405 | . . . . . 6 ⊢ ↾t Fn (V × V) | |
11 | 10 | fndmi 6658 | . . . . 5 ⊢ dom ↾t = (V × V) |
12 | 11 | releqi 5779 | . . . 4 ⊢ (Rel dom ↾t ↔ Rel (V × V)) |
13 | 9, 12 | mpbir 230 | . . 3 ⊢ Rel dom ↾t |
14 | 13 | ovprc2 7460 | . 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅) |
15 | 8, 14 | pm2.61i 182 | 1 ⊢ (∅ ↾t 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∩ cin 3946 ∅c0 4323 ↦ cmpt 5231 × cxp 5676 dom cdm 5678 ran crn 5679 Rel wrel 5683 (class class class)co 7420 ↾t crest 17401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-rest 17403 |
This theorem is referenced by: firest 17413 topnval 17415 resstopn 23089 ussval 24163 bj-rest00 36560 |
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