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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 6582. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
Ref | Expression |
---|---|
reldmress | ⊢ Rel dom ↾s |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ress 15702 | . 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑎 ∩ (Base‘𝑤))〉))) | |
2 | 1 | reldmmpt2 6669 | 1 ⊢ Rel dom ↾s |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ifcif 4036 〈cop 4131 dom cdm 5038 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 ndxcnx 15692 sSet csts 15693 Basecbs 15695 ↾s cress 15696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 df-oprab 6553 df-mpt2 6554 df-ress 15702 |
This theorem is referenced by: ressbas 15757 ressbasss 15759 resslem 15760 ress0 15761 ressinbas 15763 ressress 15765 wunress 15767 subcmn 18065 srasca 19002 rlmsca2 19022 resstopn 20800 cphsubrglem 22785 submomnd 29041 suborng 29146 |
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