Theorem feq123 5313

Description: Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)

Assertion

Ref	Expression
feq123	|- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )

Proof of Theorem feq123

Step	Hyp	Ref	Expression
1		simp1 982	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> F = G )
2		simp2 983	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> A = C )
3		simp3 984	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> B = D )
4	1, 2, 3	feq123d 5312	1 |- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 963   = wceq 1335   -->wf 5168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-fun 5174  df-fn 5175  df-f 5176

This theorem is referenced by: (None)