New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  fss GIF version

Theorem fss 5230
 Description: Expanding the codomain of a mapping. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 10-May-1998.) (Revised by set.mm contributors, 18-Sep-2011.)
Assertion
Ref Expression
fss ((F:A–→B B C) → F:A–→C)

Proof of Theorem fss
StepHypRef Expression
1 sstr2 3279 . . . . 5 (ran F B → (B C → ran F C))
21com12 27 . . . 4 (B C → (ran F B → ran F C))
32anim2d 548 . . 3 (B C → ((F Fn A ran F B) → (F Fn A ran F C)))
4 df-f 4791 . . 3 (F:A–→B ↔ (F Fn A ran F B))
5 df-f 4791 . . 3 (F:A–→C ↔ (F Fn A ran F C))
63, 4, 53imtr4g 261 . 2 (B C → (F:A–→BF:A–→C))
76impcom 419 1 ((F:A–→B B C) → F:A–→C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ⊆ wss 3257  ran crn 4773   Fn wfn 4776  –→wf 4777 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  f1ss  5262  ffoss  5314  fsn2  5434  map0  6025  mapsn  6026  mapss  6027
 Copyright terms: Public domain W3C validator