Theorem reseq1 2575

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 1638	. 2 |- (A = B -> (A i^i (C X. V)) = (B i^i (C X. V)))
2		df-res 2430	. 2 |- (A |` C) = (A i^i (C X. V))
3		df-res 2430	. 2 |- (B |` C) = (B i^i (C X. V))
4	1, 2, 3	3eqtr4g 1147	1 |- (A = B -> (A |` C) = (B |` C))

Syntax hints:   -> wi 2   = wceq 1091  Vcvv 1348   i^i cin 1486   X. cxp 2408   |` cres 2412

This theorem is referenced by:  imaeq1 2602  fun2ssres 2699  funimacnv 2711  tfrlem3 2951  tfrlem12 2960  f1stres 3096  mapunen 3397  facnnt 4870  fac0 4871  ruclem6 4890  ruclem7 4891

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-res 2430

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