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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 4823 | . . . 4 ⊢ ∅ ∈ V | |
2 | restval 16134 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴))) | |
3 | 1, 2 | mpan 706 | . . 3 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴))) |
4 | mpt0 6059 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅ | |
5 | 4 | rneqi 5384 | . . . 4 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ran ∅ |
6 | rn0 5409 | . . . 4 ⊢ ran ∅ = ∅ | |
7 | 5, 6 | eqtri 2673 | . . 3 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅ |
8 | 3, 7 | syl6eq 2701 | . 2 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅) |
9 | relxp 5160 | . . . 4 ⊢ Rel (V × V) | |
10 | restfn 16132 | . . . . . 6 ⊢ ↾t Fn (V × V) | |
11 | fndm 6028 | . . . . . 6 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V)) | |
12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ dom ↾t = (V × V) |
13 | 12 | releqi 5236 | . . . 4 ⊢ (Rel dom ↾t ↔ Rel (V × V)) |
14 | 9, 13 | mpbir 221 | . . 3 ⊢ Rel dom ↾t |
15 | 14 | ovprc2 6725 | . 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅) |
16 | 8, 15 | pm2.61i 176 | 1 ⊢ (∅ ↾t 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∩ cin 3606 ∅c0 3948 ↦ cmpt 4762 × cxp 5141 dom cdm 5143 ran crn 5144 Rel wrel 5148 Fn wfn 5921 (class class class)co 6690 ↾t crest 16128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-rest 16130 |
This theorem is referenced by: firest 16140 topnval 16142 resstopn 21038 ussval 22110 bj-rest00 33159 |
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