Theorem feq3 5213

Description: Equality theorem for functions. (Contributed by set.mm contributors, 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 3294	. . 3 |- (A = B -> (ran F C_ A <-> ran F C_ B))
2	1	anbi2d 684	. 2 |- (A = B -> ((F Fn C /\ ran F C_ A) <-> (F Fn C /\ ran F C_ B)))
3		df-f 4792	. 2 |- (F:C-->A <-> (F Fn C /\ ran F C_ A))
4		df-f 4792	. 2 |- (F:C-->B <-> (F Fn C /\ ran F C_ B))
5	2, 3, 4	3bitr4g 279	1 |- (A = B -> (F:C-->A <-> F:C-->B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   C_ wss 3258  ran crn 4774   Fn wfn 4777  -->wf 4778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-f 4792

This theorem is referenced by:  feq23  5214  fconstg  5252  f1eq3  5256  fsng  5434  fsn2  5435  mapex  6007  mapvalg  6010  mapsn  6027