Theorem reseq1 3368

Description: Equality theorem for restrictions.

Assertion

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

Proof of Theorem reseq1

Step	Hyp	Ref	Expression
1		ineq1 2210	. 2 |- (A = B -> (A i^i (C X. V)) = (B i^i (C X. V)))
2		df-res 3190	. 2 |- (A |` C) = (A i^i (C X. V))
3		df-res 3190	. 2 |- (B |` C) = (B i^i (C X. V))
4	1, 2, 3	3eqtr4g 1531	1 |- (A = B -> (A |` C) = (B |` C))

Syntax hints:   -> wi 3   = wceq 956  Vcvv 1811   i^i cin 2046   X. cxp 3168   |` cres 3172

This theorem is referenced by:  imaeq1 3401  fun2ssres 3553  cnvresid 3569  funcnvres2 3570  fvsnun1 3795  fvsnun2 3796  tfrlem3 3913  tfrlem12 3922  resoprab 4009  f1stres 4093  f2ndres 4094  mapunen 4502  seq0fval 6535  seqzfval 6537  seq1seqz 6541  seq0seqz 6542  cbvsum 6986  efseq0ex 7311  reeff1 7410  ruclem6 7515  idcn 7766  dfrelog 8756  relogf1o 8757  h2hlm 8850  ghomgrplem 10389

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-res 3190

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