Theorem reldmress 16296

Description: The structure restriction is a proper operator, so it can be used with ovprc1 6948. (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 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

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