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Theorem f11o 7113
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 7112 . . 3 (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
32anbi1i 730 . 2 ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ↔ (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
4 df-f1 5881 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
5 dff1o3 6130 . . . . . 6 (𝐹:𝐴1-1-onto𝑥 ↔ (𝐹:𝐴onto𝑥 ∧ Fun 𝐹))
65anbi1i 730 . . . . 5 ((𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ ((𝐹:𝐴onto𝑥 ∧ Fun 𝐹) ∧ 𝑥𝐵))
7 an32 838 . . . . 5 (((𝐹:𝐴onto𝑥 ∧ Fun 𝐹) ∧ 𝑥𝐵) ↔ ((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
86, 7bitri 264 . . . 4 ((𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ ((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
98exbii 1772 . . 3 (∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ ∃𝑥((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
10 19.41v 1912 . . 3 (∃𝑥((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹) ↔ (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
119, 10bitri 264 . 2 (∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
123, 4, 113bitr4i 292 1 (𝐹:𝐴1-1𝐵 ↔ ∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1702  wcel 1988  Vcvv 3195  wss 3567  ccnv 5103  Fun wfun 5870  wf 5872  1-1wf1 5873  ontowfo 5874  1-1-ontowf1o 5875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-cnv 5112  df-dm 5114  df-rn 5115  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883
This theorem is referenced by:  domen  7953  uspgrsprfo  41521
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