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Theorem fin 5160
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 3211 . . . 4 ((ran 𝐹𝐵 ∧ ran 𝐹𝐶) ↔ ran 𝐹 ⊆ (𝐵𝐶))
21anbi2i 445 . . 3 ((𝐹 Fn 𝐴 ∧ (ran 𝐹𝐵 ∧ ran 𝐹𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)))
3 anandi 555 . . 3 ((𝐹 Fn 𝐴 ∧ (ran 𝐹𝐵 ∧ ran 𝐹𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
42, 3bitr3i 184 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
5 df-f 4985 . 2 (𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)))
6 df-f 4985 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
7 df-f 4985 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
86, 7anbi12i 448 . 2 ((𝐹:𝐴𝐵𝐹:𝐴𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
94, 5, 83bitr4i 210 1 (𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹:𝐴𝐵𝐹:𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  cin 2987  wss 2988  ran crn 4412   Fn wfn 4976  wf 4977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-ss 3001  df-f 4985
This theorem is referenced by: (None)
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