Theorem fof 5270

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 3324	. . 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 4794	. 2 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
4		df-f 4792	. 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 3258  ran crn 4774   Fn wfn 4777  -->wf 4778  -onto->wfo 4780

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  df-fo 4794

This theorem is referenced by:  fofun  5271  fofn  5272  dffo2  5274  foima  5275  resdif  5307  ffoss  5315  dffo4  5424  fconst5  5456  fconstfv  5457  mapsn  6027  xpassen  6058