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Theorem dff1o3 5380
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))

Proof of Theorem dff1o3
StepHypRef Expression
1 3anan32 974 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun 𝐹))
2 dff1o2 5379 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
3 df-fo 5136 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
43anbi1i 454 . 2 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun 𝐹))
51, 2, 43bitr4i 211 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  w3a 963   = wceq 1332  ccnv 4545  ran crn 4547  Fun wfun 5124   Fn wfn 5125  ontowfo 5128  1-1-ontowf1o 5129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3081  df-ss 3088  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137
This theorem is referenced by:  f1ofo  5381  resdif  5396  f11o  5407  f1opw  5984  1stconst  6125  2ndconst  6126  f1o2ndf1  6132  ssdomg  6679  phplem4  6756  phplem4on  6768
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