Theorem reldmress 17175

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7448. (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 17174	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7543	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949  ifcif 4529  ⟨cop 4635  dom cdm 5677  Rel wrel 5682  ‘cfv 6544  (class class class)co 7409   sSet csts 17096  ndxcnx 17126  Basecbs 17144   ↾_s cress 17173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dm 5687  df-oprab 7413  df-mpo 7414  df-ress 17174

This theorem is referenced by:  ressbas  17179  ressbasOLD  17180  ressbasssg  17181  ressbasssOLD  17184  resseqnbas  17186  resslemOLD  17187  ress0  17188  ressinbas  17190  ressress  17193  wunress  17195  wunressOLD  17196  subcmn  19705  srasca  20798  srascaOLD  20799  rlmsca2  20823  resstopn  22690  cphsubrglem  24694  submomnd  32228  suborng  32433