Theorem reldmress 17293

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7451. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Assertion

Ref	Expression
reldmress	⊢ Rel dom ↾s

Proof of Theorem reldmress

Dummy variables 𝑤 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ress 17292	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7546	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906  ifcif 4488  ⟨cop 4596  dom cdm 5663  Rel wrel 5668  ‘cfv 6538  (class class class)co 7412   sSet csts 17224  ndxcnx 17254  Basecbs 17270   ↾_s cress 17291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-dm 5673  df-oprab 7416  df-mpo 7417  df-ress 17292

This theorem is referenced by:  ressbas  17297  ressbasssg  17298  ressbasssOLD  17301  resseqnbas  17303  ress0  17304  ressinbas  17306  ressress  17308  wunress  17310  subcmn  19908  submomnd  20203  suborng  20960  srasca  21282  rlmsca2  21301  resstopn  23324  cphsubrglem  25317