Theorem fin 6744

Description: Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-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 4190	. . . 4 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶) ↔ ran 𝐹 ⊆ (𝐵 ∩ 𝐶))
2	1	anbi2i 632	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)))
3		anandi 686	. . 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 279	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)))
5		df-f 6525	. 2 ⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)))
6		df-f 6525	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
7		df-f 6525	. . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
8	6, 7	anbi12i 637	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴⟶𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)))
9	4, 5, 8	3bitr4i 305	1 ⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴⟶𝐶))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∩ cin 3903   ⊆ wss 3904  ran crn 5648   Fn wfn 6516  ⟶wf 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911  df-ss 3921  df-f 6525

This theorem is referenced by:  umgrislfupgr  29321  usgrislfuspgr  29385  maprnin  32930  reprinrn  34909  reprinfz1  34913  inmap  45782