Theorem feq123 6035

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 1061	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> F = G )
2		simp2 1062	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> A = C )
3		simp3 1063	. 2 |- ( ( F = G /\ A = C /\ B = D ) -> B = D )
4	1, 2, 3	feq123d 6034	1 |- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  feq12i  6038  hashfxnn0  13124  mbfresfi  33456