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Theorem f1osetex 8884
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 8883 . 2 {𝑓𝑓:𝐴onto𝐵} ∈ V
2 f1ofo 6851 . . 3 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
32ss2abi 4063 . 2 {𝑓𝑓:𝐴1-1-onto𝐵} ⊆ {𝑓𝑓:𝐴onto𝐵}
41, 3ssexi 5326 1 {𝑓𝑓:𝐴1-1-onto𝐵} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  {cab 2705  Vcvv 3473  ontowfo 6551  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-10 2129  ax-11 2146  ax-12 2166  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-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  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-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853
This theorem is referenced by:  hashfacen  14453  symgplusg  19344  symgvalstruct  19358  symgvalstructOLD  19359
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