Theorem fofn 5552

Description: An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)

Assertion

Ref	Expression
fofn	⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)

Proof of Theorem fofn

Step	Hyp	Ref	Expression
1		fof 5550	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5473	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   Fn wfn 5313  ⟶wf 5314  –onto→wfo 5316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-f 5322  df-fo 5324

This theorem is referenced by:  fodmrnu  5558  foun  5593  fo00  5611  foelcdmi  5688  foima2  5881  cbvfo  5915  cbvexfo  5916  foeqcnvco  5920  canth  5958  1stcof  6315  2ndcof  6316  1stexg  6319  2ndexg  6320  df1st2  6371  df2nd2  6372  1stconst  6373  2ndconst  6374  fidcenumlemrks  7131  fidcenumlemr  7133  ctm  7287  suplocexprlemell  7911  ennnfonelemhf1o  12999  ennnfonelemrn  13005  imasaddfnlemg  13362  imasmnd2  13500  imasgrp2  13662  imasrng  13934  imasring  14042  znf1o  14630  upxp  14961  uptx  14963  cnmpt1st  14977  cnmpt2nd  14978  pw1nct  16428