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Theorem f1osetex 8855
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 8854 . 2 {𝑓𝑓:𝐴onto𝐵} ∈ V
2 f1ofo 6834 . . 3 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
32ss2abi 4058 . 2 {𝑓𝑓:𝐴1-1-onto𝐵} ⊆ {𝑓𝑓:𝐴onto𝐵}
41, 3ssexi 5315 1 {𝑓𝑓:𝐴1-1-onto𝐵} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  {cab 2703  Vcvv 3468  ontowfo 6535  1-1-ontowf1o 6536
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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824
This theorem is referenced by:  hashfacen  14419  symgplusg  19302  symgvalstruct  19316  symgvalstructOLD  19317
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