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Theorem fin 5998
 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 3797 . . . 4 ((ran 𝐹𝐵 ∧ ran 𝐹𝐶) ↔ ran 𝐹 ⊆ (𝐵𝐶))
21anbi2i 726 . . 3 ((𝐹 Fn 𝐴 ∧ (ran 𝐹𝐵 ∧ ran 𝐹𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)))
3 anandi 867 . . 3 ((𝐹 Fn 𝐴 ∧ (ran 𝐹𝐵 ∧ ran 𝐹𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
42, 3bitr3i 265 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
5 df-f 5808 . 2 (𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)))
6 df-f 5808 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
7 df-f 5808 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
86, 7anbi12i 729 . 2 ((𝐹:𝐴𝐵𝐹:𝐴𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
94, 5, 83bitr4i 291 1 (𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹:𝐴𝐵𝐹:𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∩ cin 3539   ⊆ wss 3540  ran crn 5039   Fn wfn 5799  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-f 5808 This theorem is referenced by:  umgrislfupgr  25789  maprnin  28894  inmap  38396  usgrislfuspgr  40414
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