Theorem feq3 6028

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 3627	. . 3 |- ( A = B -> ( ran F C_ A <-> ran F C_ B ) )
2	1	anbi2d 740	. 2 |- ( A = B -> ( ( F Fn C /\ ran F C_ A ) <-> ( F Fn C /\ ran F C_ B ) ) )
3		df-f 5892	. 2 |- ( F : C --> A <-> ( F Fn C /\ ran F C_ A ) )
4		df-f 5892	. 2 |- ( F : C --> B <-> ( F Fn C /\ ran F C_ B ) )
5	2, 3, 4	3bitr4g 303	1 |- ( A = B -> ( F : C --> A <-> F : C --> B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574   ran crn 5115   Fn wfn 5883   -->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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-f 5892

This theorem is referenced by:  feq23  6029  feq3d  6032  fun2  6067  fconstg  6092  f1eq3  6098  mapvalg  7867  mapsn  7899  cantnff  8571  axdc4uz  12783  supcvg  14588  lmff  21105  txcn  21429  lmmbr  23056  iscmet3  23091  dvcnvrelem2  23781  itgsubstlem  23811  umgrislfupgr  26018  usgrislfuspgr  26079  wlkv0  26547  isgrpo  27351  vciOLD  27416  isvclem  27432  nmop0h  28850  sitgaddlemb  30410  sitmcl  30413  cvmliftlem15  31280  mtyf  31449  matunitlindflem1  33405  sdclem1  33539  k0004lem1  38445  mapsnd  39388  stoweidlem57  40274