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Theorem fin 5246
 Description: Mapping into an intersection. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Sep-1999.) (Revised by set.mm contributors, 18-Sep-2011.)
Assertion
Ref Expression
fin (F:A–→(BC) ↔ (F:A–→B F:A–→C))

Proof of Theorem fin
StepHypRef Expression
1 ssin 3477 . . . 4 ((ran F B ran F C) ↔ ran F (BC))
21anbi2i 675 . . 3 ((F Fn A (ran F B ran F C)) ↔ (F Fn A ran F (BC)))
3 anandi 801 . . 3 ((F Fn A (ran F B ran F C)) ↔ ((F Fn A ran F B) (F Fn A ran F C)))
42, 3bitr3i 242 . 2 ((F Fn A ran F (BC)) ↔ ((F Fn A ran F B) (F Fn A ran F C)))
5 df-f 4791 . 2 (F:A–→(BC) ↔ (F Fn A ran F (BC)))
6 df-f 4791 . . 3 (F:A–→B ↔ (F Fn A ran F B))
7 df-f 4791 . . 3 (F:A–→C ↔ (F Fn A ran F C))
86, 7anbi12i 678 . 2 ((F:A–→B F:A–→C) ↔ ((F Fn A ran F B) (F Fn A ran F C)))
94, 5, 83bitr4i 268 1 (F:A–→(BC) ↔ (F:A–→B F:A–→C))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∩ cin 3208   ⊆ wss 3257  ran crn 4773   Fn wfn 4776  –→wf 4777 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791 This theorem is referenced by: (None)
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