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Theorem dff1o5 5295
 Description: Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 22-Oct-2011.)
Assertion
Ref Expression
dff1o5 (F:A1-1-ontoB ↔ (F:A1-1B ran F = B))

Proof of Theorem dff1o5
StepHypRef Expression
1 df-f1o 4794 . 2 (F:A1-1-ontoB ↔ (F:A1-1B F:AontoB))
2 f1f 5258 . . . . 5 (F:A1-1BF:A–→B)
32biantrurd 494 . . . 4 (F:A1-1B → (ran F = B ↔ (F:A–→B ran F = B)))
4 dffo2 5273 . . . 4 (F:AontoB ↔ (F:A–→B ran F = B))
53, 4syl6rbbr 255 . . 3 (F:A1-1B → (F:AontoB ↔ ran F = B))
65pm5.32i 618 . 2 ((F:A1-1B F:AontoB) ↔ (F:A1-1B ran F = B))
71, 6bitri 240 1 (F:A1-1-ontoB ↔ (F:A1-1B ran F = B))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642  ran crn 4773  –→wf 4777  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 4780 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  df-f1 4792  df-fo 4793  df-f1o 4794 This theorem is referenced by:  f1orescnv  5301
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