Theorem reseq2 5058

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 4788	. . 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 4786	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4786	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
5	2, 3, 4	3eqtr4g 2296	1 |- ( A = B -> ( C |` A ) = ( C |` B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   _Vcvv 2821   i^i cin 3219   X. cxp 4772   |` cres 4776

This proof depends on 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 proof 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 4193  df-xp 4780  df-res 4786

This theorem is used by:  reseq2i  5060  reseq2d  5063  resabs1  5092  resima2  5097  imaeq2  5122  resdisj  5216  relcoi1  5319  fressnfv  5902  tfrlem1  6579  tfrlem9  6590  tfr0dm  6593  tfrlemisucaccv  6596  tfrlemiubacc  6601  tfr1onlemsucaccv  6612  tfr1onlemubacc  6617  tfr1onlemaccex  6619  tfrcllemsucaccv  6625  tfrcllembxssdm  6627  tfrcllemubacc  6630  tfrcllemaccex  6632  tfrcllemres  6633  tfrcldm  6634  fnfi  7250  gsumclfi  14159  gsummptfidmadd  14161  gsumsubmclfi  14163  lmbr2  15315  lmff  15350  dvmptid  15817