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Theorem f1oexbi 7942
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 3477 . . . . 5 𝑓 ∈ V
21cnvex 7939 . . . 4 𝑓 ∈ V
3 f1ocnv 6856 . . . 4 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
4 f1oeq1 6832 . . . . 5 (𝑔 = 𝑓 → (𝑔:𝐵1-1-onto𝐴𝑓:𝐵1-1-onto𝐴))
54spcegv 3586 . . . 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 1925 . 2 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → ∃𝑔 𝑔:𝐵1-1-onto𝐴)
8 vex 3477 . . . . 5 𝑔 ∈ V
98cnvex 7939 . . . 4 𝑔 ∈ V
10 f1ocnv 6856 . . . 4 (𝑔:𝐵1-1-onto𝐴𝑔:𝐴1-1-onto𝐵)
11 f1oeq1 6832 . . . . 5 (𝑓 = 𝑔 → (𝑓:𝐴1-1-onto𝐵𝑔:𝐴1-1-onto𝐵))
1211spcegv 3586 . . . 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 1925 . 2 (∃𝑔 𝑔:𝐵1-1-onto𝐴 → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
157, 14impbii 208 1 (∃𝑓 𝑓:𝐴1-1-onto𝐵 ↔ ∃𝑔 𝑔:𝐵1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wex 1773  wcel 2098  Vcvv 3473  ccnv 5681  1-1-ontowf1o 6552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560
This theorem is referenced by:  rusgrnumwlkg  29808  f1ocnt  32591
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