Theorem reseq2 4770

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 4511	. . 3 |- ( A = B -> ( A X. _V ) = ( B X. _V ) )
2	1	ineq2d 3241	. 2 |- ( A = B -> ( C i^i ( A X. _V ) ) = ( C i^i ( B X. _V ) ) )
3		df-res 4509	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4509	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
5	2, 3, 4	3eqtr4g 2170	1 |- ( A = B -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1312   _Vcvv 2655   i^i cin 3034   X. cxp 4495   |` cres 4499

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-in 3041  df-opab 3948  df-xp 4503  df-res 4509

This theorem is referenced by:  reseq2i  4772  reseq2d  4775  resabs1  4804  resima2  4809  imaeq2  4833  resdisj  4923  relcoi1  5026  fressnfv  5559  tfrlem1  6156  tfrlem9  6167  tfr0dm  6170  tfrlemisucaccv  6173  tfrlemiubacc  6178  tfr1onlemsucaccv  6189  tfr1onlemubacc  6194  tfr1onlemaccex  6196  tfrcllemsucaccv  6202  tfrcllembxssdm  6204  tfrcllemubacc  6207  tfrcllemaccex  6209  tfrcllemres  6210  tfrcldm  6211  fnfi  6774  lmbr2  12218  lmff  12253