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Theorem f1oexbi 7158
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 3234 . . . . 5 𝑓 ∈ V
21cnvex 7155 . . . 4 𝑓 ∈ V
3 f1ocnv 6187 . . . 4 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
4 f1oeq1 6165 . . . . 5 (𝑔 = 𝑓 → (𝑔:𝐵1-1-onto𝐴𝑓:𝐵1-1-onto𝐴))
54spcegv 3325 . . . 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 1898 . 2 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → ∃𝑔 𝑔:𝐵1-1-onto𝐴)
8 vex 3234 . . . . 5 𝑔 ∈ V
98cnvex 7155 . . . 4 𝑔 ∈ V
10 f1ocnv 6187 . . . 4 (𝑔:𝐵1-1-onto𝐴𝑔:𝐴1-1-onto𝐵)
11 f1oeq1 6165 . . . . 5 (𝑓 = 𝑔 → (𝑓:𝐴1-1-onto𝐵𝑔:𝐴1-1-onto𝐵))
1211spcegv 3325 . . . 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 1898 . 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 1744  wcel 2030  Vcvv 3231  ccnv 5142  1-1-ontowf1o 5925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933
This theorem is referenced by:  rusgrnumwlkg  26944  f1ocnt  29687
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