Theorem dffn3 4996

Description: A function maps to its range. (Contributed by NM, 1-Sep-1999.)

Assertion

Ref	Expression
dffn3	⊢ (𝐹 Fn A ↔ 𝐹:A⟶ran 𝐹)

Proof of Theorem dffn3

Step	Hyp	Ref	Expression
1		ssid 2958	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2	1	biantru 286	. 2 ⊢ (𝐹 Fn A ↔ (𝐹 Fn A ∧ ran 𝐹 ⊆ ran 𝐹))
3		df-f 4849	. 2 ⊢ (𝐹:A⟶ran 𝐹 ↔ (𝐹 Fn A ∧ ran 𝐹 ⊆ ran 𝐹))
4	2, 3	bitr4i 176	1 ⊢ (𝐹 Fn A ↔ 𝐹:A⟶ran 𝐹)

Syntax hints:   ∧ wa 97   ↔ wb 98   ⊆ wss 2911  ran crn 4289   Fn wfn 4840  ⟶wf 4841

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-f 4849

This theorem is referenced by:  fsn2  5280  fo2ndf  5790