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Theorem fof 5269
 Description: An onto mapping is a mapping. (Contributed by set.mm contributors, 3-Aug-1994.)
Assertion
Ref Expression
fof (F:AontoBF:A–→B)

Proof of Theorem fof
StepHypRef Expression
1 eqimss 3323 . . 3 (ran F = B → ran F B)
21anim2i 552 . 2 ((F Fn A ran F = B) → (F Fn A ran F B))
3 df-fo 4793 . 2 (F:AontoB ↔ (F Fn A ran F = B))
4 df-f 4791 . 2 (F:A–→B ↔ (F Fn A ran F B))
52, 3, 43imtr4i 257 1 (F:AontoBF:A–→B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3257  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779 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  df-fo 4793 This theorem is referenced by:  fofun  5270  fofn  5271  dffo2  5273  foima  5274  resdif  5306  ffoss  5314  dffo4  5423  fconst5  5455  fconstfv  5456  mapsn  6026  xpassen  6057
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