Theorem ssres 3377

Description: Subclass theorem for restriction.

Assertion

Ref	Expression
ssres	|- (A (_ B -> (A |` C) (_ (B |` C))

Proof of Theorem ssres

Step	Hyp	Ref	Expression
1		ssrin 2230	. 2 |- (A (_ B -> (A i^i (C X. V)) (_ (B i^i (C X. V)))
2		df-res 3185	. 2 |- (A |` C) = (A i^i (C X. V))
3		df-res 3185	. 2 |- (B |` C) = (B i^i (C X. V))
4	1, 2, 3	3sstr4g 2098	1 |- (A (_ B -> (A |` C) (_ (B |` C))

Syntax hints:   -> wi 3  Vcvv 1807   i^i cin 2042   (_ wss 2043   X. cxp 3163   |` cres 3167

This theorem is referenced by:  imass1 3424  sspg 8334  ssps 8336  sspn 8342

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-in 2047  df-ss 2049  df-res 3185

Copyright terms: Public domain