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Theorem fin 5441
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 3382 . . . 4 ((ran 𝐹𝐵 ∧ ran 𝐹𝐶) ↔ ran 𝐹 ⊆ (𝐵𝐶))
21anbi2i 457 . . 3 ((𝐹 Fn 𝐴 ∧ (ran 𝐹𝐵 ∧ ran 𝐹𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)))
3 anandi 590 . . 3 ((𝐹 Fn 𝐴 ∧ (ran 𝐹𝐵 ∧ ran 𝐹𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
42, 3bitr3i 186 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
5 df-f 5259 . 2 (𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵𝐶)))
6 df-f 5259 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
7 df-f 5259 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
86, 7anbi12i 460 . 2 ((𝐹:𝐴𝐵𝐹:𝐴𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
94, 5, 83bitr4i 212 1 (𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹:𝐴𝐵𝐹:𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  cin 3153  wss 3154  ran crn 4661   Fn wfn 5250  wf 5251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-ss 3167  df-f 5259
This theorem is referenced by: (None)
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