Theorem reseq2 4955

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 4690	. . 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 4688	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4688	. 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 4674   |` cres 4678

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 4107  df-xp 4682  df-res 4688

This theorem is referenced by:  reseq2i  4957  reseq2d  4960  resabs1  4989  resima2  4994  imaeq2  5019  resdisj  5112  relcoi1  5215  fressnfv  5773  tfrlem1  6396  tfrlem9  6407  tfr0dm  6410  tfrlemisucaccv  6413  tfrlemiubacc  6418  tfr1onlemsucaccv  6429  tfr1onlemubacc  6434  tfr1onlemaccex  6436  tfrcllemsucaccv  6442  tfrcllembxssdm  6444  tfrcllemubacc  6447  tfrcllemaccex  6449  tfrcllemres  6450  tfrcldm  6451  fnfi  7040  lmbr2  14719  lmff  14754  dvmptid  15221