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Theorem f1oexbi 7662
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 3403 . . . . 5 𝑓 ∈ V
21cnvex 7659 . . . 4 𝑓 ∈ V
3 f1ocnv 6633 . . . 4 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
4 f1oeq1 6609 . . . . 5 (𝑔 = 𝑓 → (𝑔:𝐵1-1-onto𝐴𝑓:𝐵1-1-onto𝐴))
54spcegv 3502 . . . 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 1937 . 2 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → ∃𝑔 𝑔:𝐵1-1-onto𝐴)
8 vex 3403 . . . . 5 𝑔 ∈ V
98cnvex 7659 . . . 4 𝑔 ∈ V
10 f1ocnv 6633 . . . 4 (𝑔:𝐵1-1-onto𝐴𝑔:𝐴1-1-onto𝐵)
11 f1oeq1 6609 . . . . 5 (𝑓 = 𝑔 → (𝑓:𝐴1-1-onto𝐵𝑔:𝐴1-1-onto𝐵))
1211spcegv 3502 . . . 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 1937 . 2 (∃𝑔 𝑔:𝐵1-1-onto𝐴 → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
157, 14impbii 212 1 (∃𝑓 𝑓:𝐴1-1-onto𝐵 ↔ ∃𝑔 𝑔:𝐵1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wex 1786  wcel 2114  Vcvv 3399  ccnv 5525  1-1-ontowf1o 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347
This theorem is referenced by:  rusgrnumwlkg  27918  f1ocnt  30701
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