Theorem f1osetex 8906

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⟶1-1onto𝐵} ∈ V

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem f1osetex

Step	Hyp	Ref	Expression
1		fosetex 8905	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶onto𝐵} ∈ V
2		f1ofo 6851	. . 3 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝑓:𝐴⟶onto𝐵)
3	2	ss2abi 3994	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴⟶onto𝐵}
4	1, 3	ssexi 5288	1 ⊢ {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐵} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2147  {cab 2741  Vcvv 3454  ⟶ontowfo 6553  ⟶1-1ontowf1o 6554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-map 8874

This theorem is used by:  hashfacen  14623  symgplusg  19624  symgvalstruct  19638