Theorem permsetexOLD 18892

Description: Obsolete version of f1osetex 8605 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 8579	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {𝑓 ∣ 𝑓:𝐴⟶𝐴} ∈ V)
2	1	anidms 566	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐴} ∈ V)
3		f1of 6700	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐴 → 𝑓:𝐴⟶𝐴)
4	3	ss2abi 3996	. . 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 5243	1 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  {cab 2715  Vcvv 3422   ⊆ wss 3883  ⟶wf 6414  –1-1-onto→wf1o 6417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-f1o 6425

This theorem is referenced by:  symgbasexOLD  18894