Theorem fofn 5248

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 5246	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5174	. 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 5023  ⟶wf 5024  –onto→wfo 5026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3006  df-ss 3013  df-f 5032  df-fo 5034

This theorem is referenced by:  fodmrnu  5254  foun  5285  fo00  5302  foima2  5544  cbvfo  5578  cbvexfo  5579  foeqcnvco  5583  1stcof  5948  2ndcof  5949  1stexg  5952  2ndexg  5953  df1st2  5998  df2nd2  5999  1stconst  6000  2ndconst  6001  fidcenumlemrks  6716  fidcenumlemr  6718  ctm  6845