Theorem fnf 2753

Description: Any function is a mapping into V.

Assertion

Ref	Expression
fnf	⊢ (F Fn A ↔ F:A–→V)

Proof of Theorem fnf

Step	Hyp	Ref	Expression
1		ssv 1520	. . 3 ⊢ ran F ⊆ V
2		df-f 2434	. . . 4 ⊢ (F:A–→VF Fn A ∧ ran F ⊆ V))
3	2	biimpr 134	. . 3 ⊢ ((F Fn A ∧ ran F ⊆ V) → F:A–→V)
4	1, 3	mpan2 519	. 2 ⊢ (F Fn A → F:A–→V)
5		ffn 2752	. 2 ⊢ (F:A–→V → F Fn A)
6	4, 5	impbi 139	1 ⊢ (F Fn A ↔ F:A–→V)

Syntax hints:   ↔ wb 127   ∧ wa 196  Vcvv 1348   ⊆ wss 1487  ran crn 2411   Fn wfn 2417  –→wf 2418

This theorem is referenced by:  f1cnv 2782  fnressn 2897  tz7.48lem 2993

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-v 1349  df-in 1491  df-ss 1492  df-f 2434

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