Theorem fofn 5620

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 5618	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5536	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)

This proof depends on syntax axioms:   → wi 4   Fn wfn 5375  ⟶wf 5376  –onto→wfo 5378

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-f 5384  df-fo 5386

This theorem is used by:  fodmrnu  5626  foun  5661  fo00  5680  foelcdmi  5758  foima2  5960  cbvfo  5994  cbvexfo  5995  foeqcnvco  5999  canth  6039  1stcof  6401  2ndcof  6402  1stexg  6405  2ndexg  6406  df1st2  6459  df2nd2  6460  1stconst  6461  2ndconst  6462  fidcenumlemrks  7275  fidcenumlemr  7277  ctm  7455  suplocexprlemell  8086  ennnfonelemhf1o  13362  ennnfonelemrn  13368  imasaddfnlemg  13694  imasmnd2  13818  imasgrp2  13972  imasrng  14362  imasring  14478  znf1o  15095  upxp  15498  uptx  15500  cnmpt1st  15514  cnmpt2nd  15515  pw1nct  17238