Theorem f1osetex 8798

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 8797	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ∈ V
2		f1ofo 6780	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
3	2	ss2abi 4017	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–onto→𝐵}
4	1, 3	ssexi 5266	1 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ∈ V

Syntax hints:   ∈ wcel 2114  {cab 2713  Vcvv 3439  –onto→wfo 6489  –1-1-onto→wf1o 6490

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8767

This theorem is referenced by:  hashfacen  14379  symgplusg  19314  symgvalstruct  19328