Theorem reseq1 4675

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 3183	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4423	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4423	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2142	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1287   _Vcvv 2615   i^i cin 2987   X. cxp 4409   |` cres 4413

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-res 4423

This theorem is referenced by:  reseq1i  4677  reseq1d  4680  imaeq1  4736  relcoi1  4928  tfr0dm  6041  tfrlemiex  6050  tfr1onlemex  6066  tfr1onlemaccex  6067  tfrcllemsucaccv  6073  tfrcllembxssdm  6075  tfrcllemex  6079  tfrcllemaccex  6080  tfrcllemres  6081  pmresg  6385