Theorem feq123 5038

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 904	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> F = G )
2		simp2 905	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> A = C )
3		simp3 906	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> B = D )
4	1, 2, 3	feq123d 5037	1 |- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )

Syntax hints:   -> wi 4   <-> wb 98   /\ w3a 885   = wceq 1243   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906

This theorem is referenced by: (None)