Theorem fin 2770

Description: Function into an intersection.

Assertion

Ref	Expression
fin	⊢ (F:A–→(B ∩ C) ↔ (F:A–→B ∧ F:A–→C))

Proof of Theorem fin

Step	Hyp	Ref	Expression
1		anidm 331	. . . 4 ⊢ ((F Fn A ∧ F Fn A) ↔ F Fn A)
2		ssin 1659	. . . 4 ⊢ ((ran F ⊆ B ∧ ran F ⊆ C) ↔ ran F ⊆ (B ∩ C))
3	1, 2	anbi12i 369	. . 3 ⊢ (((F Fn A ∧ F Fn A) ∧ (ran F ⊆ B ∧ ran F ⊆ C)) ↔ (F Fn A ∧ ran F ⊆ (B ∩ C)))
4		an4 388	. . 3 ⊢ (((F Fn A ∧ F Fn A) ∧ (ran F ⊆ B ∧ ran F ⊆ C)) ↔ ((F Fn A ∧ ran F ⊆ B) ∧ (F Fn A ∧ ran F ⊆ C)))
5	3, 4	bitr3 153	. 2 ⊢ ((F Fn A ∧ ran F ⊆ (B ∩ C)) ↔ ((F Fn A ∧ ran F ⊆ B) ∧ (F Fn A ∧ ran F ⊆ C)))
6		df-f 2434	. 2 ⊢ (F:A–→(B ∩ C) ↔ (F Fn A ∧ ran F ⊆ (B ∩ C)))
7		df-f 2434	. . 3 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
8		df-f 2434	. . 3 ⊢ (F:A–→C ↔ (F Fn A ∧ ran F ⊆ C))
9	7, 8	anbi12i 369	. 2 ⊢ ((F:A–→B ∧ F:A–→C) ↔ ((F Fn A ∧ ran F ⊆ B) ∧ (F Fn A ∧ ran F ⊆ C)))
10	5, 6, 9	3bitr4 158	1 ⊢ (F:A–→(B ∩ C) ↔ (F:A–→B ∧ F:A–→C))

Syntax hints:   ↔ wb 127   ∧ wa 196   ∩ cin 1486   ⊆ wss 1487  ran crn 2411   Fn wfn 2417  –→wf 2418

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

metamath.org