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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 17471 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴))) | |
| 3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴))) |
| 4 | mpt0 6710 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅ | |
| 5 | 4 | rneqi 5948 | . . . 4 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ran ∅ |
| 6 | rn0 5936 | . . . 4 ⊢ ran ∅ = ∅ | |
| 7 | 5, 6 | eqtri 2765 | . . 3 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅ |
| 8 | 3, 7 | eqtrdi 2793 | . 2 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅) |
| 9 | relxp 5703 | . . . 4 ⊢ Rel (V × V) | |
| 10 | restfn 17469 | . . . . . 6 ⊢ ↾t Fn (V × V) | |
| 11 | 10 | fndmi 6672 | . . . . 5 ⊢ dom ↾t = (V × V) |
| 12 | 11 | releqi 5787 | . . . 4 ⊢ (Rel dom ↾t ↔ Rel (V × V)) |
| 13 | 9, 12 | mpbir 231 | . . 3 ⊢ Rel dom ↾t |
| 14 | 13 | ovprc2 7471 | . 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅) |
| 15 | 8, 14 | pm2.61i 182 | 1 ⊢ (∅ ↾t 𝐴) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ∅c0 4333 ↦ cmpt 5225 × cxp 5683 dom cdm 5685 ran crn 5686 Rel wrel 5690 (class class class)co 7431 ↾t crest 17465 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-rest 17467 |
| This theorem is referenced by: firest 17477 topnval 17479 resstopn 23194 ussval 24268 bj-rest00 37082 |
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