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Theorem dff1o5 6044
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 5797 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 f1f 5999 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
32biantrurd 527 . . . 4 (𝐹:𝐴1-1𝐵 → (ran 𝐹 = 𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵)))
4 dffo2 6017 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
53, 4syl6rbbr 277 . . 3 (𝐹:𝐴1-1𝐵 → (𝐹:𝐴onto𝐵 ↔ ran 𝐹 = 𝐵))
65pm5.32i 666 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
71, 6bitri 262 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  ran crn 5029  wf 5786  1-1wf1 5787  ontowfo 5788  1-1-ontowf1o 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-in 3546  df-ss 3553  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797
This theorem is referenced by:  f1orescnv  6050  domdifsn  7905  sucdom2  8018  ackbij1  8920  ackbij2  8925  fin4en1  8991  om2uzf1oi  12569  s4f1o  13459  fvcosymgeq  17618  indlcim  19940  2lgslem1b  24834  ausisusgra  25650  usgraexmpledg  25698  cdleme50f1o  34648  diaf1oN  35233  pwssplit4  36473  meadjiunlem  39155  ausgrusgrb  40390  usgrexmpledg  40481
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