Theorem reseq1 5057

Description: Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)

Assertion

Ref	Expression
reseq1	|- ( A = B -> ( A |` C ) = ( B |` C ) )

Proof of Theorem reseq1

Step	Hyp	Ref	Expression
1		ineq1 3425	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4786	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4786	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2296	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

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-res 4786

This theorem is used by:  reseq1i  5059  reseq1d  5062  imaeq1  5121  relcoi1  5319  tfr0dm  6593  tfrlemiex  6602  tfr1onlemex  6618  tfr1onlemaccex  6619  tfrcllemsucaccv  6625  tfrcllembxssdm  6627  tfrcllemex  6631  tfrcllemaccex  6632  tfrcllemres  6633  pmresg  6957  mapunen  7151  hashf1lem1  11285  lmbr  15314