Theorem feq3 5322

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 3166	. . 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 5192	. 2 |- ( F : C --> A <-> ( F Fn C /\ ran F C_ A ) )
4		df-f 5192	. 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 1343   C_ wss 3116   ran crn 4605   Fn wfn 5183   -->wf 5184

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129  df-f 5192

This theorem is referenced by:  feq23  5323  feq3d  5326  feq123d  5328  fun2  5361  fconstg  5384  f1eq3  5390  fsng  5658  fsn2  5659  fsnunf  5685  mapvalg  6624  mapsn  6656  lmff  12899  txcn  12925