Theorem reldmress 17200

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7402. (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 17199	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7497	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890  ifcif 4461  ⟨cop 4568  dom cdm 5625  Rel wrel 5630  ‘cfv 6492  (class class class)co 7363   sSet csts 17131  ndxcnx 17161  Basecbs 17177   ↾_s cress 17198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-dm 5635  df-oprab 7367  df-mpo 7368  df-ress 17199

This theorem is referenced by:  ressbas  17204  ressbasssg  17205  ressbasssOLD  17208  resseqnbas  17210  ress0  17211  ressinbas  17213  ressress  17215  wunress  17217  subcmn  19810  submomnd  20105  suborng  20855  srasca  21177  rlmsca2  21196  resstopn  23176  cphsubrglem  25169