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Theorem f1oexbi 7078
Description: There is a one-to-one onto function from a set to a second set iff there is a one-to-one onto function from the second set to the first set. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
Assertion
Ref Expression
f1oexbi (∃𝑓 𝑓:𝐴1-1-onto𝐵 ↔ ∃𝑔 𝑔:𝐵1-1-onto𝐴)
Distinct variable groups:   𝐴,𝑓,𝑔   𝐵,𝑓,𝑔

Proof of Theorem f1oexbi
StepHypRef Expression
1 vex 3193 . . . . 5 𝑓 ∈ V
21cnvex 7075 . . . 4 𝑓 ∈ V
3 f1ocnv 6116 . . . 4 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
4 f1oeq1 6094 . . . . 5 (𝑔 = 𝑓 → (𝑔:𝐵1-1-onto𝐴𝑓:𝐵1-1-onto𝐴))
54spcegv 3284 . . . 4 (𝑓 ∈ V → (𝑓:𝐵1-1-onto𝐴 → ∃𝑔 𝑔:𝐵1-1-onto𝐴))
62, 3, 5mpsyl 68 . . 3 (𝑓:𝐴1-1-onto𝐵 → ∃𝑔 𝑔:𝐵1-1-onto𝐴)
76exlimiv 1855 . 2 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → ∃𝑔 𝑔:𝐵1-1-onto𝐴)
8 vex 3193 . . . . 5 𝑔 ∈ V
98cnvex 7075 . . . 4 𝑔 ∈ V
10 f1ocnv 6116 . . . 4 (𝑔:𝐵1-1-onto𝐴𝑔:𝐴1-1-onto𝐵)
11 f1oeq1 6094 . . . . 5 (𝑓 = 𝑔 → (𝑓:𝐴1-1-onto𝐵𝑔:𝐴1-1-onto𝐵))
1211spcegv 3284 . . . 4 (𝑔 ∈ V → (𝑔:𝐴1-1-onto𝐵 → ∃𝑓 𝑓:𝐴1-1-onto𝐵))
139, 10, 12mpsyl 68 . . 3 (𝑔:𝐵1-1-onto𝐴 → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
1413exlimiv 1855 . 2 (∃𝑔 𝑔:𝐵1-1-onto𝐴 → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
157, 14impbii 199 1 (∃𝑓 𝑓:𝐴1-1-onto𝐵 ↔ ∃𝑔 𝑔:𝐵1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wex 1701  wcel 1987  Vcvv 3190  ccnv 5083  1-1-ontowf1o 5856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864
This theorem is referenced by:  rusgrnumwlkg  26773  f1ocnt  29442
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