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Theorem fin 5980
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 3793 . . . 4 ((ran 𝐹𝐵 ∧ ran 𝐹𝐶) ↔ ran 𝐹 ⊆ (𝐵𝐶))
21anbi2i 725 . . 3 ((𝐹 Fn 𝐴 ∧ (ran 𝐹𝐵 ∧ ran 𝐹𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)))
3 anandi 866 . . 3 ((𝐹 Fn 𝐴 ∧ (ran 𝐹𝐵 ∧ ran 𝐹𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
42, 3bitr3i 264 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
5 df-f 5791 . 2 (𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)))
6 df-f 5791 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
7 df-f 5791 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
86, 7anbi12i 728 . 2 ((𝐹:𝐴𝐵𝐹:𝐴𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
94, 5, 83bitr4i 290 1 (𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹:𝐴𝐵𝐹:𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  cin 3535  wss 3536  ran crn 5026   Fn wfn 5782  wf 5783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-v 3171  df-in 3543  df-ss 3550  df-f 5791
This theorem is referenced by:  maprnin  28697  inmap  38196  umgrislfupgr  40347  usgrislfuspgr  40413
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