Theorem reseq2 5053

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 4783	. . 3 |- ( A = B -> ( A X. _V ) = ( B X. _V ) )
2	1	ineq2d 3432	. 2 |- ( A = B -> ( C i^i ( A X. _V ) ) = ( C i^i ( B X. _V ) ) )
3		df-res 4781	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4781	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
5	2, 3, 4	3eqtr4g 2296	1 |- ( A = B -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1402   _Vcvv 2821   i^i cin 3219   X. cxp 4767   |` cres 4771

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-opab 4188  df-xp 4775  df-res 4781

This theorem is referenced by:  reseq2i  5055  reseq2d  5058  resabs1  5087  resima2  5092  imaeq2  5117  resdisj  5211  relcoi1  5314  fressnfv  5893  tfrlem1  6569  tfrlem9  6580  tfr0dm  6583  tfrlemisucaccv  6586  tfrlemiubacc  6591  tfr1onlemsucaccv  6602  tfr1onlemubacc  6607  tfr1onlemaccex  6609  tfrcllemsucaccv  6615  tfrcllembxssdm  6617  tfrcllemubacc  6620  tfrcllemaccex  6622  tfrcllemres  6623  tfrcldm  6624  fnfi  7240  gsumclfi  14136  gsummptfidmadd  14138  gsumsubmclfi  14140  lmbr2  15238  lmff  15273  dvmptid  15740