Theorem reseq1 3374

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 2213	. 2 |- (A = B -> (A i^i (C X. V)) = (B i^i (C X. V)))
2		df-res 3196	. 2 |- (A |` C) = (A i^i (C X. V))
3		df-res 3196	. 2 |- (B |` C) = (B i^i (C X. V))
4	1, 2, 3	3eqtr4g 1534	1 |- (A = B -> (A |` C) = (B |` C))

Syntax hints:   -> wi 3   = wceq 958  Vcvv 1814   i^i cin 2049   X. cxp 3174   |` cres 3178

This theorem is referenced by:  imaeq1 3407  fun2ssres 3559  cnvresid 3575  funcnvres2 3576  fvsnun1 3801  fvsnun2 3802  tfrlem3 3919  tfrlem12 3928  resoprab 4015  f1stres 4099  f2ndres 4100  mapunen 4508  seq0fval 6536  seqzfval 6538  seq1seqz 6542  seq0seqz 6543  cbvsum 6986  efseq0ex 7311  reeff1 7410  ruclem6 7516  idcn 7763  dfrelog 8751  relogf1o 8752  h2hlm 8845  ghomgrplem 10384

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-res 3196

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