Theorem fofn 5412

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 5410	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5337	. 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 5183  ⟶wf 5184  –onto→wfo 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129  df-f 5192  df-fo 5194

This theorem is referenced by:  fodmrnu  5418  foun  5451  fo00  5468  foima2  5720  cbvfo  5753  cbvexfo  5754  foeqcnvco  5758  canth  5796  1stcof  6131  2ndcof  6132  1stexg  6135  2ndexg  6136  df1st2  6187  df2nd2  6188  1stconst  6189  2ndconst  6190  fidcenumlemrks  6918  fidcenumlemr  6920  ctm  7074  suplocexprlemell  7654  ennnfonelemhf1o  12346  ennnfonelemrn  12352  upxp  12912  uptx  12914  cnmpt1st  12928  cnmpt2nd  12929  pw1nct  13883