Theorem permaxsep 45248

Description: The Axiom of Separation ax-sep 5241 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148.

Note that, to prove that an instance of Separation holds in the model, 𝜑 would need have all instances of ∈ replaced with 𝑅. But this still results in an instance of this theorem, so we do establish that Separation holds. (Contributed by Eric Schmidt, 6-Nov-2025.)

Hypotheses

Ref	Expression
permmodel.1	⊢ 𝐹:V–1-1-onto→V
permmodel.2	⊢ 𝑅 = (◡𝐹 ∘ E )

Assertion

Ref	Expression
permaxsep	⊢ ∃𝑦∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧   𝑥,𝐹,𝑦   𝑦,𝑅

Allowed substitution hints:   𝜑(𝑥)   𝑅(𝑥,𝑧)   𝐹(𝑧)

Proof of Theorem permaxsep

Step	Hyp	Ref	Expression
1		fvex 6847	. 2 ⊢ (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ∈ V
2		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥◡𝐹
3		nfrab1 3419	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}
4	2, 3	nffv 6844	. . . 4 ⊢ Ⅎ𝑥(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
5	4	nfeq2 2916	. . 3 ⊢ Ⅎ𝑥 𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
6		breq2 5102	. . . 4 ⊢ (𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) → (𝑥𝑅𝑦 ↔ 𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})))
7	6	bibi1d 343	. . 3 ⊢ (𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) → ((𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑)) ↔ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))))
8	5, 7	albid 2229	. 2 ⊢ (𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) → (∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑)) ↔ ∀𝑥(𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))))
9		permmodel.1	. . . . 5 ⊢ 𝐹:V–1-1-onto→V
10		permmodel.2	. . . . 5 ⊢ 𝑅 = (◡𝐹 ∘ E )
11		vex 3444	. . . . 5 ⊢ 𝑥 ∈ V
12		fvex 6847	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
13	12	rabex 5284	. . . . 5 ⊢ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ∈ V
14	9, 10, 11, 13	brpermmodelcnv 45245	. . . 4 ⊢ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ 𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
15		rabid 3420	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ↔ (𝑥 ∈ (𝐹‘𝑧) ∧ 𝜑))
16		vex 3444	. . . . . . 7 ⊢ 𝑧 ∈ V
17	9, 10, 11, 16	brpermmodel 45244	. . . . . 6 ⊢ (𝑥𝑅𝑧 ↔ 𝑥 ∈ (𝐹‘𝑧))
18	17	bicomi 224	. . . . 5 ⊢ (𝑥 ∈ (𝐹‘𝑧) ↔ 𝑥𝑅𝑧)
19	15, 18	bianbi 627	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ↔ (𝑥𝑅𝑧 ∧ 𝜑))
20	14, 19	bitri 275	. . 3 ⊢ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))
21	20	ax-gen 1796	. 2 ⊢ ∀𝑥(𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))
22	1, 8, 21	ceqsexv2d 3491	1 ⊢ ∃𝑦∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {crab 3399  Vcvv 3440   class class class wbr 5098   E cep 5523  ^◡ccnv 5623   ∘ ccom 5628  –1-1-onto→wf1o 6491  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by: (None)