Theorem fofn 5558

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 5556	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5479	. 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 5319  ⟶wf 5320  –onto→wfo 5322

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 3204  df-ss 3211  df-f 5328  df-fo 5330

This theorem is referenced by:  fodmrnu  5564  foun  5599  fo00  5617  foelcdmi  5694  foima2  5887  cbvfo  5921  cbvexfo  5922  foeqcnvco  5926  canth  5964  1stcof  6321  2ndcof  6322  1stexg  6325  2ndexg  6326  df1st2  6379  df2nd2  6380  1stconst  6381  2ndconst  6382  fidcenumlemrks  7143  fidcenumlemr  7145  ctm  7299  suplocexprlemell  7923  ennnfonelemhf1o  13024  ennnfonelemrn  13030  imasaddfnlemg  13387  imasmnd2  13525  imasgrp2  13687  imasrng  13959  imasring  14067  znf1o  14655  upxp  14986  uptx  14988  cnmpt1st  15002  cnmpt2nd  15003  pw1nct  16540