Theorem feq123 5376

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 999	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> F = G )
2		simp2 1000	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> A = C )
3		simp3 1001	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> B = D )
4	1, 2, 3	feq123d 5375	1 |- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1364   -->wf 5231

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-f 5239

This theorem is referenced by: (None)