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Theorem dff1o5 6794
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 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 6504 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 dffo2 6761 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
3 f1f 6739 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
43biantrurd 534 . . . 4 (𝐹:𝐴1-1𝐵 → (ran 𝐹 = 𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵)))
52, 4bitr4id 290 . . 3 (𝐹:𝐴1-1𝐵 → (𝐹:𝐴onto𝐵 ↔ ran 𝐹 = 𝐵))
65pm5.32i 576 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
71, 6bitri 275 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  ran crn 5635  wf 6493  1-1wf1 6494  ontowfo 6495  1-1-ontowf1o 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3448  df-in 3918  df-ss 3928  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by:  f1orescnv  6800  domdifsn  8999  sucdom2OLD  9027  sucdom2  9151  ackbij1  10175  ackbij2  10180  fin4en1  10246  om2uzf1oi  13859  s4f1o  14808  fvcosymgeq  19212  indlcim  21249  2lgslem1b  26743  ausgrusgrb  28119  usgrexmpledg  28213  cdleme50f1o  39012  diaf1oN  39596  pwssplit4  41419  cantnf2  41662  meadjiunlem  44713
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