Theorem reseq2 4954

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 4689	. . 3 |- ( A = B -> ( A X. _V ) = ( B X. _V ) )
2	1	ineq2d 3374	. 2 |- ( A = B -> ( C i^i ( A X. _V ) ) = ( C i^i ( B X. _V ) ) )
3		df-res 4687	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4687	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
5	2, 3, 4	3eqtr4g 2263	1 |- ( A = B -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1373   _Vcvv 2772   i^i cin 3165   X. cxp 4673   |` cres 4677

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-opab 4106  df-xp 4681  df-res 4687

This theorem is referenced by:  reseq2i  4956  reseq2d  4959  resabs1  4988  resima2  4993  imaeq2  5018  resdisj  5111  relcoi1  5214  fressnfv  5771  tfrlem1  6394  tfrlem9  6405  tfr0dm  6408  tfrlemisucaccv  6411  tfrlemiubacc  6416  tfr1onlemsucaccv  6427  tfr1onlemubacc  6432  tfr1onlemaccex  6434  tfrcllemsucaccv  6440  tfrcllembxssdm  6442  tfrcllemubacc  6445  tfrcllemaccex  6447  tfrcllemres  6448  tfrcldm  6449  fnfi  7038  lmbr2  14686  lmff  14721  dvmptid  15188