Theorem reseq2 4904

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 4642	. . 3 |- ( A = B -> ( A X. _V ) = ( B X. _V ) )
2	1	ineq2d 3338	. 2 |- ( A = B -> ( C i^i ( A X. _V ) ) = ( C i^i ( B X. _V ) ) )
3		df-res 4640	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4640	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
5	2, 3, 4	3eqtr4g 2235	1 |- ( A = B -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1353   _Vcvv 2739   i^i cin 3130   X. cxp 4626   |` cres 4630

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-opab 4067  df-xp 4634  df-res 4640

This theorem is referenced by:  reseq2i  4906  reseq2d  4909  resabs1  4938  resima2  4943  imaeq2  4968  resdisj  5059  relcoi1  5162  fressnfv  5705  tfrlem1  6311  tfrlem9  6322  tfr0dm  6325  tfrlemisucaccv  6328  tfrlemiubacc  6333  tfr1onlemsucaccv  6344  tfr1onlemubacc  6349  tfr1onlemaccex  6351  tfrcllemsucaccv  6357  tfrcllembxssdm  6359  tfrcllemubacc  6362  tfrcllemaccex  6364  tfrcllemres  6365  tfrcldm  6366  fnfi  6938  lmbr2  13753  lmff  13788