Theorem fofn 5441

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 5439	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5366	. 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 5212  ⟶wf 5213  –onto→wfo 5215

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3136  df-ss 3143  df-f 5221  df-fo 5223

This theorem is referenced by:  fodmrnu  5447  foun  5481  fo00  5498  foelcdmi  5569  foima2  5753  cbvfo  5786  cbvexfo  5787  foeqcnvco  5791  canth  5829  1stcof  6164  2ndcof  6165  1stexg  6168  2ndexg  6169  df1st2  6220  df2nd2  6221  1stconst  6222  2ndconst  6223  fidcenumlemrks  6952  fidcenumlemr  6954  ctm  7108  suplocexprlemell  7712  ennnfonelemhf1o  12414  ennnfonelemrn  12420  imasaddfnlemg  12735  upxp  13775  uptx  13777  cnmpt1st  13791  cnmpt2nd  13792  pw1nct  14755