Theorem fin 5247

Description: Mapping into an intersection. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Sep-1999.) (Revised by set.mm contributors, 18-Sep-2011.)

Assertion

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

Proof of Theorem fin

Step	Hyp	Ref	Expression
1		ssin 3478	. . . 4 ⊢ ((ran F ⊆ B ∧ ran F ⊆ C) ↔ ran F ⊆ (B ∩ C))
2	1	anbi2i 675	. . 3 ⊢ ((F Fn A ∧ (ran F ⊆ B ∧ ran F ⊆ C)) ↔ (F Fn A ∧ ran F ⊆ (B ∩ C)))
3		anandi 801	. . 3 ⊢ ((F Fn A ∧ (ran F ⊆ B ∧ ran F ⊆ C)) ↔ ((F Fn A ∧ ran F ⊆ B) ∧ (F Fn A ∧ ran F ⊆ C)))
4	2, 3	bitr3i 242	. 2 ⊢ ((F Fn A ∧ ran F ⊆ (B ∩ C)) ↔ ((F Fn A ∧ ran F ⊆ B) ∧ (F Fn A ∧ ran F ⊆ C)))
5		df-f 4792	. 2 ⊢ (F:A–→(B ∩ C) ↔ (F Fn A ∧ ran F ⊆ (B ∩ C)))
6		df-f 4792	. . 3 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
7		df-f 4792	. . 3 ⊢ (F:A–→C ↔ (F Fn A ∧ ran F ⊆ C))
8	6, 7	anbi12i 678	. 2 ⊢ ((F:A–→B ∧ F:A–→C) ↔ ((F Fn A ∧ ran F ⊆ B) ∧ (F Fn A ∧ ran F ⊆ C)))
9	4, 5, 8	3bitr4i 268	1 ⊢ (F:A–→(B ∩ C) ↔ (F:A–→B ∧ F:A–→C))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   ∩ cin 3209   ⊆ wss 3258  ran crn 4774   Fn wfn 4777  –→wf 4778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-f 4792

This theorem is used by: (None)