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Theorem f1osetex 8850
Description: The set of bijections between two classes exists. (Contributed by AV, 30-Mar-2024.) (Revised by AV, 8-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
Assertion
Ref Expression
f1osetex {𝑓𝑓:𝐴1-1-onto𝐵} ∈ V
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem f1osetex
StepHypRef Expression
1 fosetex 8849 . 2 {𝑓𝑓:𝐴onto𝐵} ∈ V
2 f1ofo 6831 . . 3 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
32ss2abi 4056 . 2 {𝑓𝑓:𝐴1-1-onto𝐵} ⊆ {𝑓𝑓:𝐴onto𝐵}
41, 3ssexi 5313 1 {𝑓𝑓:𝐴1-1-onto𝐵} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  {cab 2701  Vcvv 3466  ontowfo 6532  1-1-ontowf1o 6533
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819
This theorem is referenced by:  hashfacen  14411  symgplusg  19294  symgvalstruct  19308  symgvalstructOLD  19309
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