Theorem feq123 5359

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 997	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> F = G )
2		simp2 998	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> A = C )
3		simp3 999	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> B = D )
4	1, 2, 3	feq123d 5358	1 |- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   = wceq 1353   -->wf 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222

This theorem is referenced by: (None)