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Theorem permsetex 18498
Description: The set of permutations of a set 𝐴 exists. (Contributed by AV, 30-Mar-2024.)
Assertion
Ref Expression
permsetex (𝐴𝑉 → {𝑓𝑓:𝐴1-1-onto𝐴} ∈ V)
Distinct variable group:   𝐴,𝑓
Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem permsetex
StepHypRef Expression
1 mapex 8408 . . 3 ((𝐴𝑉𝐴𝑉) → {𝑓𝑓:𝐴𝐴} ∈ V)
21anidms 570 . 2 (𝐴𝑉 → {𝑓𝑓:𝐴𝐴} ∈ V)
3 f1of 6606 . . . 4 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴𝐴)
43ss2abi 4029 . . 3 {𝑓𝑓:𝐴1-1-onto𝐴} ⊆ {𝑓𝑓:𝐴𝐴}
54a1i 11 . 2 (𝐴𝑉 → {𝑓𝑓:𝐴1-1-onto𝐴} ⊆ {𝑓𝑓:𝐴𝐴})
62, 5ssexd 5214 1 (𝐴𝑉 → {𝑓𝑓:𝐴1-1-onto𝐴} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  {cab 2802  Vcvv 3480  wss 3919  wf 6339  1-1-ontowf1o 6342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-f1o 6350
This theorem is referenced by:  symgbasex  18500  symgplusg  18511
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