Theorem fofn 5456

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 5454	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5381	. 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 5227  ⟶wf 5228  –onto→wfo 5230

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 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-f 5236  df-fo 5238

This theorem is referenced by:  fodmrnu  5462  foun  5496  fo00  5513  foelcdmi  5585  foima2  5769  cbvfo  5803  cbvexfo  5804  foeqcnvco  5808  canth  5846  1stcof  6183  2ndcof  6184  1stexg  6187  2ndexg  6188  df1st2  6239  df2nd2  6240  1stconst  6241  2ndconst  6242  fidcenumlemrks  6977  fidcenumlemr  6979  ctm  7133  suplocexprlemell  7737  ennnfonelemhf1o  12459  ennnfonelemrn  12465  imasaddfnlemg  12784  imasgrp2  13045  imasrng  13303  imasring  13407  upxp  14209  uptx  14211  cnmpt1st  14225  cnmpt2nd  14226  pw1nct  15190