Theorem fof 5353

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

Assertion

Ref	Expression
fof	⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)

Proof of Theorem fof

Step	Hyp	Ref	Expression
1		eqimss 3156	. . 3 ⊢ (ran 𝐹 = 𝐵 → ran 𝐹 ⊆ 𝐵)
2	1	anim2i 340	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
3		df-fo 5137	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4		df-f 5135	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
5	2, 3, 4	3imtr4i 200	1 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ⊆ wss 3076  ran crn 4548   Fn wfn 5126  ⟶wf 5127  –onto→wfo 5129

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089  df-f 5135  df-fo 5137

This theorem is referenced by:  fofun  5354  fofn  5355  dffo2  5357  foima  5358  resdif  5397  ffoss  5407  fconstfvm  5646  cocan2  5697  foeqcnvco  5699  fornex  6021  algrflem  6134  algrflemg  6135  tposf2  6173  mapsn  6592  ssdomg  6680  fopwdom  6738  fidcenumlemrks  6849  fidcenumlemr  6851  ctmlemr  7001  ctm  7002  ctssdclemn0  7003  ctssdccl  7004  ctssdc  7006  enumctlemm  7007  enumct  7008  fodjuomnilemdc  7024  exmidfodomrlemr  7075  exmidfodomrlemrALT  7076  suplocexprlemdisj  7552  suplocexprlemub  7555  focdmex  10565  ennnfonelemdc  11948  ennnfonelemg  11952  ennnfonelemp1  11955  ennnfonelemhdmp1  11958  ennnfonelemkh  11961  ennnfonelemhf1o  11962  ennnfonelemex  11963  ennnfonelemhom  11964  ctinfomlemom  11976  ctinf  11979  ctiunctlemudc  11986  ctiunctlemf  11987  omctfn  11992  dvrecap  12885  subctctexmid  13369  pw1nct  13371