Theorem fof 5269

Description: An onto mapping is a mapping. (Contributed by set.mm contributors, 3-Aug-1994.)

Assertion

Ref	Expression
fof	|- (F:A-onto->B -> F:A-->B)

Proof of Theorem fof

Step	Hyp	Ref	Expression
1		eqimss 3323	. . 3 |- (ran F = B -> ran F C_ B)
2	1	anim2i 552	. 2 |- ((F Fn A /\ ran F = B) -> (F Fn A /\ ran F C_ B))
3		df-fo 4793	. 2 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
4		df-f 4791	. 2 |- (F:A-->B <-> (F Fn A /\ ran F C_ B))
5	2, 3, 4	3imtr4i 257	1 |- (F:A-onto->B -> F:A-->B)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   C_ wss 3257  ran crn 4773   Fn wfn 4776  -->wf 4777  -onto->wfo 4779

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791  df-fo 4793

This theorem is referenced by:  fofun  5270  fofn  5271  dffo2  5273  foima  5274  resdif  5306  ffoss  5314  dffo4  5423  fconst5  5455  fconstfv  5456  mapsn  6026  xpassen  6057