| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > reldmress | Structured version Visualization version GIF version | ||
| Description: The structure restriction is a proper operator, so it can be used with ovprc1 7470. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
| Ref | Expression |
|---|---|
| reldmress | ⊢ Rel dom ↾s |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ress 17275 | . 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑎 ∩ (Base‘𝑤))〉))) | |
| 2 | 1 | reldmmpo 7567 | 1 ⊢ Rel dom ↾s |
| Colors of variables: wff setvar class |
| Syntax hints: Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ifcif 4525 〈cop 4632 dom cdm 5685 Rel wrel 5690 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 ndxcnx 17230 Basecbs 17247 ↾s cress 17274 |
| 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-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 df-oprab 7435 df-mpo 7436 df-ress 17275 |
| This theorem is referenced by: ressbas 17280 ressbasOLD 17281 ressbasssg 17282 ressbasssOLD 17285 resseqnbas 17287 resslemOLD 17288 ress0 17289 ressinbas 17291 ressress 17293 wunress 17295 wunressOLD 17296 subcmn 19855 srasca 21183 srascaOLD 21184 rlmsca2 21206 resstopn 23194 cphsubrglem 25211 submomnd 33087 suborng 33345 |
| Copyright terms: Public domain | W3C validator |