Theorem fofn 5485

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 5483	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5410	. 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 5254  ⟶wf 5255  –onto→wfo 5257

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-f 5263  df-fo 5265

This theorem is referenced by:  fodmrnu  5491  foun  5526  fo00  5543  foelcdmi  5616  foima2  5801  cbvfo  5835  cbvexfo  5836  foeqcnvco  5840  canth  5878  1stcof  6230  2ndcof  6231  1stexg  6234  2ndexg  6235  df1st2  6286  df2nd2  6287  1stconst  6288  2ndconst  6289  fidcenumlemrks  7028  fidcenumlemr  7030  ctm  7184  suplocexprlemell  7797  ennnfonelemhf1o  12655  ennnfonelemrn  12661  imasaddfnlemg  13016  imasmnd2  13154  imasgrp2  13316  imasrng  13588  imasring  13696  znf1o  14283  upxp  14592  uptx  14594  cnmpt1st  14608  cnmpt2nd  14609  pw1nct  15734