Theorem reseq2 5033

Description: Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)

Assertion

Ref	Expression
reseq2	|- ( A = B -> ( C |` A ) = ( C |` B ) )

Proof of Theorem reseq2

Step	Hyp	Ref	Expression
1		xpeq1 4763	. . 3 |- ( A = B -> ( A X. _V ) = ( B X. _V ) )
2	1	ineq2d 3422	. 2 |- ( A = B -> ( C i^i ( A X. _V ) ) = ( C i^i ( B X. _V ) ) )
3		df-res 4761	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4761	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
5	2, 3, 4	3eqtr4g 2290	1 |- ( A = B -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1398   _Vcvv 2813   i^i cin 3210   X. cxp 4747   |` cres 4751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-opab 4172  df-xp 4755  df-res 4761

This theorem is referenced by:  reseq2i  5035  reseq2d  5038  resabs1  5067  resima2  5072  imaeq2  5097  resdisj  5191  relcoi1  5294  fressnfv  5871  tfrlem1  6539  tfrlem9  6550  tfr0dm  6553  tfrlemisucaccv  6556  tfrlemiubacc  6561  tfr1onlemsucaccv  6572  tfr1onlemubacc  6577  tfr1onlemaccex  6579  tfrcllemsucaccv  6585  tfrcllembxssdm  6587  tfrcllemubacc  6590  tfrcllemaccex  6592  tfrcllemres  6593  tfrcldm  6594  fnfi  7203  lmbr2  15079  lmff  15114  dvmptid  15581  gfsumcl  16870