Theorem permsetexOLD 18734

Description: Obsolete version of f1osetex 8529 as of 8-Aug-2024. (Contributed by AV, 30-Mar-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
permsetexOLD	⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V)

Distinct variable group:   𝐴,𝑓

Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem permsetexOLD

Step	Hyp	Ref	Expression
1		mapex 8503	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {𝑓 ∣ 𝑓:𝐴⟶𝐴} ∈ V)
2	1	anidms 570	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐴} ∈ V)
3		f1of 6650	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐴 → 𝑓:𝐴⟶𝐴)
4	3	ss2abi 3970	. . 3 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐴}
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐴})
6	2, 5	ssexd 5206	1 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2110  {cab 2712  Vcvv 3401   ⊆ wss 3857  ⟶wf 6365  –1-1-onto→wf1o 6368

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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-br 5044  df-opab 5106  df-xp 5546  df-rel 5547  df-cnv 5548  df-dm 5550  df-rn 5551  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-f1o 6376

This theorem is referenced by:  symgbasexOLD  18736