Theorem fof 5420

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 3201	. . 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 5204	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4		df-f 5202	. 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 1348   ⊆ wss 3121  ran crn 4612   Fn wfn 5193  ⟶wf 5194  –onto→wfo 5196

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-f 5202  df-fo 5204

This theorem is referenced by:  fofun  5421  fofn  5422  dffo2  5424  foima  5425  resdif  5464  ffoss  5474  fconstfvm  5714  cocan2  5767  foeqcnvco  5769  fornex  6094  algrflem  6208  algrflemg  6209  tposf2  6247  mapsn  6668  ssdomg  6756  fopwdom  6814  fidcenumlemrks  6930  fidcenumlemr  6932  ctmlemr  7085  ctm  7086  ctssdclemn0  7087  ctssdccl  7088  ctssdc  7090  enumctlemm  7091  enumct  7092  fodjuomnilemdc  7120  exmidfodomrlemr  7179  exmidfodomrlemrALT  7180  suplocexprlemdisj  7682  suplocexprlemub  7685  focdmex  10721  ennnfonelemdc  12354  ennnfonelemg  12358  ennnfonelemp1  12361  ennnfonelemhdmp1  12364  ennnfonelemkh  12367  ennnfonelemhf1o  12368  ennnfonelemex  12369  ennnfonelemhom  12370  ctinfomlemom  12382  ctinf  12385  ctiunctlemudc  12392  ctiunctlemf  12393  omctfn  12398  dvrecap  13471  subctctexmid  14034  pw1nct  14036