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  5892  cbvfo  5926  cbvexfo  5927  foeqcnvco  5931  canth  5969  1stcof  6326  2ndcof  6327  1stexg  6330  2ndexg  6331  df1st2  6384  df2nd2  6385  1stconst  6386  2ndconst  6387  fidcenumlemrks  7152  fidcenumlemr  7154  ctm  7308  suplocexprlemell  7933  ennnfonelemhf1o  13039  ennnfonelemrn  13045  imasaddfnlemg  13402  imasmnd2  13540  imasgrp2  13702  imasrng  13975  imasring  14083  znf1o  14671  upxp  15002  uptx  15004  cnmpt1st  15018  cnmpt2nd  15019  pw1nct  16630