Theorem permsetexOLD 19363

Description: Obsolete version of f1osetex 8880 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 7944	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {𝑓 ∣ 𝑓:𝐴⟶𝐴} ∈ V)
2	1	anidms 565	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐴} ∈ V)
3		f1of 6835	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐴 → 𝑓:𝐴⟶𝐴)
4	3	ss2abi 4060	. . 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 5321	1 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2099  {cab 2703  Vcvv 3462   ⊆ wss 3946  ⟶wf 6542  –1-1-onto→wf1o 6545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-xp 5680  df-rel 5681  df-cnv 5682  df-dm 5684  df-rn 5685  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-f1o 6553

This theorem is referenced by:  symgbasexOLD  19365