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Theorem f11o 5513
Description: Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
Hypothesis
Ref Expression
f11o.1 𝐹 ∈ V
Assertion
Ref Expression
f11o (𝐹:𝐴1-1𝐵 ↔ ∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵

Proof of Theorem f11o
StepHypRef Expression
1 f11o.1 . . . 4 𝐹 ∈ V
21ffoss 5512 . . 3 (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
32anbi1i 458 . 2 ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ↔ (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
4 df-f1 5240 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
5 dff1o3 5486 . . . . . 6 (𝐹:𝐴1-1-onto𝑥 ↔ (𝐹:𝐴onto𝑥 ∧ Fun 𝐹))
65anbi1i 458 . . . . 5 ((𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ ((𝐹:𝐴onto𝑥 ∧ Fun 𝐹) ∧ 𝑥𝐵))
7 an32 562 . . . . 5 (((𝐹:𝐴onto𝑥 ∧ Fun 𝐹) ∧ 𝑥𝐵) ↔ ((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
86, 7bitri 184 . . . 4 ((𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ ((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
98exbii 1616 . . 3 (∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ ∃𝑥((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
10 19.41v 1914 . . 3 (∃𝑥((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹) ↔ (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
119, 10bitri 184 . 2 (∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
123, 4, 113bitr4i 212 1 (𝐹:𝐴1-1𝐵 ↔ ∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wex 1503  wcel 2160  Vcvv 2752  wss 3144  ccnv 4643  Fun wfun 5229  wf 5231  1-1wf1 5232  ontowfo 5233  1-1-ontowf1o 5234
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-cnv 4652  df-dm 4654  df-rn 4655  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242
This theorem is referenced by:  domen  6778
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