Theorem reseq2 5000

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 4733	. . 3 |- ( A = B -> ( A X. _V ) = ( B X. _V ) )
2	1	ineq2d 3405	. 2 |- ( A = B -> ( C i^i ( A X. _V ) ) = ( C i^i ( B X. _V ) ) )
3		df-res 4731	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4731	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
5	2, 3, 4	3eqtr4g 2287	1 |- ( A = B -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1395   _Vcvv 2799   i^i cin 3196   X. cxp 4717   |` cres 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-opab 4146  df-xp 4725  df-res 4731

This theorem is referenced by:  reseq2i  5002  reseq2d  5005  resabs1  5034  resima2  5039  imaeq2  5064  resdisj  5157  relcoi1  5260  fressnfv  5826  tfrlem1  6454  tfrlem9  6465  tfr0dm  6468  tfrlemisucaccv  6471  tfrlemiubacc  6476  tfr1onlemsucaccv  6487  tfr1onlemubacc  6492  tfr1onlemaccex  6494  tfrcllemsucaccv  6500  tfrcllembxssdm  6502  tfrcllemubacc  6505  tfrcllemaccex  6507  tfrcllemres  6508  tfrcldm  6509  fnfi  7103  lmbr2  14888  lmff  14923  dvmptid  15390