Theorem fofn 5549

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 5547	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5472	. 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 5312  ⟶wf 5313  –onto→wfo 5315

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 5321  df-fo 5323

This theorem is referenced by:  fodmrnu  5555  foun  5590  fo00  5608  foelcdmi  5685  foima2  5874  cbvfo  5908  cbvexfo  5909  foeqcnvco  5913  canth  5951  1stcof  6307  2ndcof  6308  1stexg  6311  2ndexg  6312  df1st2  6363  df2nd2  6364  1stconst  6365  2ndconst  6366  fidcenumlemrks  7116  fidcenumlemr  7118  ctm  7272  suplocexprlemell  7896  ennnfonelemhf1o  12979  ennnfonelemrn  12985  imasaddfnlemg  13342  imasmnd2  13480  imasgrp2  13642  imasrng  13914  imasring  14022  znf1o  14609  upxp  14940  uptx  14942  cnmpt1st  14956  cnmpt2nd  14957  pw1nct  16328