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Theorem dff1o4 5464
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
dff1o4 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))

Proof of Theorem dff1o4
StepHypRef Expression
1 dff1o2 5461 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
2 3anass 982 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun 𝐹 ∧ ran 𝐹 = 𝐵)))
3 df-rn 4633 . . . . . 6 ran 𝐹 = dom 𝐹
43eqeq1i 2185 . . . . 5 (ran 𝐹 = 𝐵 ↔ dom 𝐹 = 𝐵)
54anbi2i 457 . . . 4 ((Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))
6 df-fn 5214 . . . 4 (𝐹 Fn 𝐵 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))
75, 6bitr4i 187 . . 3 ((Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ 𝐹 Fn 𝐵)
87anbi2i 457 . 2 ((𝐹 Fn 𝐴 ∧ (Fun 𝐹 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
91, 2, 83bitri 206 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  w3a 978   = wceq 1353  ccnv 4621  dom cdm 4622  ran crn 4623  Fun wfun 5205   Fn wfn 5206  1-1-ontowf1o 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142  df-rn 4633  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218
This theorem is referenced by:  f1ocnv  5469  f1oun  5476  f1o00  5491  f1oi  5494  f1osn  5496  f1ompt  5662  f1ofveu  5856  f1ocnvd  6066  f1od2  6229  mapsnf1o2  6689  sbthlemi9  6957  mhmf1o  12738  grpinvf1o  12816  hmeof1o2  13441
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