Theorem fnfrn 2758

Description: A function maps to its range.

Assertion

Ref	Expression
fnfrn	⊢ (F Fn A ↔ F:A–→ran F)

Proof of Theorem fnfrn

Step	Hyp	Ref	Expression
1		ssid 1519	. . 3 ⊢ ran F ⊆ ran F
2	1	biantru 543	. 2 ⊢ (F Fn A ↔ (F Fn A ∧ ran F ⊆ ran F))
3		df-f 2434	. 2 ⊢ (F:A–→ran F ↔ (F Fn A ∧ ran F ⊆ ran F))
4	2, 3	bitr4 154	1 ⊢ (F Fn A ↔ F:A–→ran F)

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

This theorem is referenced by:  fsn2 2896  ac6lem 3575  fodom 3613

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

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