Theorem fofn 5561

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 5559	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5482	. 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 5321  ⟶wf 5322  –onto→wfo 5324

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213  df-f 5330  df-fo 5332

This theorem is referenced by:  fodmrnu  5567  foun  5602  fo00  5621  foelcdmi  5698  foima2  5891  cbvfo  5925  cbvexfo  5926  foeqcnvco  5930  canth  5968  1stcof  6325  2ndcof  6326  1stexg  6329  2ndexg  6330  df1st2  6383  df2nd2  6384  1stconst  6385  2ndconst  6386  fidcenumlemrks  7151  fidcenumlemr  7153  ctm  7307  suplocexprlemell  7932  ennnfonelemhf1o  13033  ennnfonelemrn  13039  imasaddfnlemg  13396  imasmnd2  13534  imasgrp2  13696  imasrng  13968  imasring  14076  znf1o  14664  upxp  14995  uptx  14997  cnmpt1st  15011  cnmpt2nd  15012  pw1nct  16604