Theorem f1osetex 8852

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 8851	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ∈ V
2		f1ofo 6840	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
3	2	ss2abi 4063	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–onto→𝐵}
4	1, 3	ssexi 5322	1 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ∈ V

Syntax hints:   ∈ wcel 2106  {cab 2709  Vcvv 3474  –onto→wfo 6541  –1-1-onto→wf1o 6542

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821

This theorem is referenced by:  hashfacen  14412  symgplusg  19249  symgvalstruct  19263  symgvalstructOLD  19264