Theorem fof 2788

Description: An onto mapping is a mapping.

Assertion

Ref	Expression
fof	⊢ (F:A–onto→B → F:A–→B)

Proof of Theorem fof

Step	Hyp	Ref	Expression
1		eqimss 1548	. . 3 ⊢ (ran F = B → ran F ⊆ B)
2	1	anim2i 270	. 2 ⊢ ((F Fn A ∧ ran F = B) → (F Fn A ∧ ran F ⊆ B))
3		df-fo 2436	. 2 ⊢ (F:A–onto→B ↔ (F Fn A ∧ ran F = B))
4		df-f 2434	. 2 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
5	2, 3, 4	3imtr4 192	1 ⊢ (F:A–onto→B → F:A–→B)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ⊆ wss 1487  ran crn 2411   Fn wfn 2417  –→wf 2418  –onto→wfo 2420

This theorem is referenced by:  foima 2790  fornex 2793  ffoss 2820  fconstfv 2903  cbvfo 2923  canth 2945  df1st2 3098  mapsn 3269  ssdomg 3311  unfilem2 3439  fiint 3445  fodom 3613  fodomb 3615  alephiso 3697  ruclem10 4894  ruclem11 4895  ruclem39 4923  infmap2lem2 4952

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-f 2434  df-fo 2436

metamath.org