Theorem f1osetex 8800

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	⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ∈ V

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem f1osetex

Step	Hyp	Ref	Expression
1		fosetex 8799	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ∈ V
2		f1ofo 6778	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
3	2	ss2abi 4000	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–onto→𝐵}
4	1, 3	ssexi 5253	1 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ∈ V

Syntax hints:   ∈ wcel 2121  {cab 2719  Vcvv 3433  –onto→wfo 6487  –1-1-onto→wf1o 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769

This theorem is referenced by:  hashfacen  14411  symgplusg  19353  symgvalstruct  19367