Theorem fof 5340

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 3146	. . 3 ⊢ (ran 𝐹 = 𝐵 → ran 𝐹 ⊆ 𝐵)
2	1	anim2i 339	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
3		df-fo 5124	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4		df-f 5122	. 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 1331   ⊆ wss 3066  ran crn 4535   Fn wfn 5113  ⟶wf 5114  –onto→wfo 5116

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079  df-f 5122  df-fo 5124

This theorem is referenced by:  fofun  5341  fofn  5342  dffo2  5344  foima  5345  resdif  5382  ffoss  5392  fconstfvm  5631  cocan2  5682  foeqcnvco  5684  fornex  6006  algrflem  6119  algrflemg  6120  tposf2  6158  mapsn  6577  ssdomg  6665  fopwdom  6723  fidcenumlemrks  6834  fidcenumlemr  6836  ctmlemr  6986  ctm  6987  ctssdclemn0  6988  ctssdccl  6989  ctssdc  6991  enumctlemm  6992  enumct  6993  fodjuomnilemdc  7009  exmidfodomrlemr  7051  exmidfodomrlemrALT  7052  suplocexprlemdisj  7521  suplocexprlemub  7524  focdmex  10526  ennnfonelemdc  11901  ennnfonelemg  11905  ennnfonelemp1  11908  ennnfonelemhdmp1  11911  ennnfonelemkh  11914  ennnfonelemhf1o  11915  ennnfonelemex  11916  ennnfonelemhom  11917  ctinfomlemom  11929  ctinf  11932  ctiunctlemudc  11939  ctiunctlemf  11940  dvrecap  12835  subctctexmid  13185