Theorem fofn 5570

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 5568	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5489	. 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 5328  ⟶wf 5329  –onto→wfo 5331

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214  df-f 5337  df-fo 5339

This theorem is referenced by:  fodmrnu  5576  foun  5611  fo00  5630  foelcdmi  5707  foima2  5902  cbvfo  5936  cbvexfo  5937  foeqcnvco  5941  canth  5979  1stcof  6335  2ndcof  6336  1stexg  6339  2ndexg  6340  df1st2  6393  df2nd2  6394  1stconst  6395  2ndconst  6396  fidcenumlemrks  7195  fidcenumlemr  7197  ctm  7351  suplocexprlemell  7976  ennnfonelemhf1o  13097  ennnfonelemrn  13103  imasaddfnlemg  13460  imasmnd2  13598  imasgrp2  13760  imasrng  14033  imasring  14141  znf1o  14730  upxp  15066  uptx  15068  cnmpt1st  15082  cnmpt2nd  15083  pw1nct  16708