Theorem fofn 5478

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 5476	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5403	. 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 5249  ⟶wf 5250  –onto→wfo 5252

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166  df-f 5258  df-fo 5260

This theorem is referenced by:  fodmrnu  5484  foun  5519  fo00  5536  foelcdmi  5609  foima2  5794  cbvfo  5828  cbvexfo  5829  foeqcnvco  5833  canth  5871  1stcof  6216  2ndcof  6217  1stexg  6220  2ndexg  6221  df1st2  6272  df2nd2  6273  1stconst  6274  2ndconst  6275  fidcenumlemrks  7012  fidcenumlemr  7014  ctm  7168  suplocexprlemell  7773  ennnfonelemhf1o  12570  ennnfonelemrn  12576  imasaddfnlemg  12897  imasgrp2  13180  imasrng  13452  imasring  13560  znf1o  14139  upxp  14440  uptx  14442  cnmpt1st  14456  cnmpt2nd  14457  pw1nct  15493