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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 6948. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
Ref | Expression |
---|---|
reldmress | ⊢ Rel dom ↾s |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ress 16237 | . 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑎 ∩ (Base‘𝑤))〉))) | |
2 | 1 | reldmmpt2 7036 | 1 ⊢ Rel dom ↾s |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3414 ∩ cin 3797 ⊆ wss 3798 ifcif 4308 〈cop 4405 dom cdm 5346 Rel wrel 5351 ‘cfv 6127 (class class class)co 6910 ndxcnx 16226 sSet csts 16227 Basecbs 16229 ↾s cress 16230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-dm 5356 df-oprab 6914 df-mpt2 6915 df-ress 16237 |
This theorem is referenced by: ressbas 16300 ressbasss 16302 resslem 16303 ress0 16304 ressinbas 16306 ressress 16309 wunress 16311 subcmn 18602 srasca 19549 rlmsca2 19569 resstopn 21368 cphsubrglem 23353 submomnd 30251 suborng 30356 |
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