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Theorem fin 6541
 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⟶(𝐵𝐶) ↔ (𝐹:𝐴𝐵𝐹:𝐴𝐶))

Proof of Theorem fin
StepHypRef Expression
1 ssin 4160 . . . 4 ((ran 𝐹𝐵 ∧ ran 𝐹𝐶) ↔ ran 𝐹 ⊆ (𝐵𝐶))
21anbi2i 625 . . 3 ((𝐹 Fn 𝐴 ∧ (ran 𝐹𝐵 ∧ ran 𝐹𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)))
3 anandi 675 . . 3 ((𝐹 Fn 𝐴 ∧ (ran 𝐹𝐵 ∧ ran 𝐹𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
42, 3bitr3i 280 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
5 df-f 6336 . 2 (𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)))
6 df-f 6336 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
7 df-f 6336 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
86, 7anbi12i 629 . 2 ((𝐹:𝐴𝐵𝐹:𝐴𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
94, 5, 83bitr4i 306 1 (𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹:𝐴𝐵𝐹:𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∩ cin 3882   ⊆ wss 3883  ran crn 5524   Fn wfn 6327  ⟶wf 6328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-in 3890  df-ss 3900  df-f 6336 This theorem is referenced by:  umgrislfupgr  26960  usgrislfuspgr  27021  maprnin  30537  reprinrn  32065  reprinfz1  32069  inmap  42008
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