Theorem feq3 5265

Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
feq3	|- ( A = B -> ( F : C --> A <-> F : C --> B ) )

Proof of Theorem feq3

Step	Hyp	Ref	Expression
1		sseq2 3126	. . 3 |- ( A = B -> ( ran F C_ A <-> ran F C_ B ) )
2	1	anbi2d 460	. 2 |- ( A = B -> ( ( F Fn C /\ ran F C_ A ) <-> ( F Fn C /\ ran F C_ B ) ) )
3		df-f 5135	. 2 |- ( F : C --> A <-> ( F Fn C /\ ran F C_ A ) )
4		df-f 5135	. 2 |- ( F : C --> B <-> ( F Fn C /\ ran F C_ B ) )
5	2, 3, 4	3bitr4g 222	1 |- ( A = B -> ( F : C --> A <-> F : C --> B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   C_ wss 3076   ran crn 4548   Fn wfn 5126   -->wf 5127

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-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089  df-f 5135

This theorem is referenced by:  feq23  5266  feq3d  5269  feq123d  5271  fun2  5304  fconstg  5327  f1eq3  5333  fsng  5601  fsn2  5602  fsnunf  5628  mapvalg  6560  mapsn  6592  lmff  12457  txcn  12483