Theorem permaxsep 44981

Description: The Axiom of Separation ax-sep 5235 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 6835	. 2 ⊢ (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ∈ V
2		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥◡𝐹
3		nfrab1 3415	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}
4	2, 3	nffv 6832	. . . 4 ⊢ Ⅎ𝑥(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
5	4	nfeq2 2909	. . 3 ⊢ Ⅎ𝑥 𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
6		breq2 5096	. . . 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 2223	. 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 3440	. . . . 5 ⊢ 𝑥 ∈ V
12		fvex 6835	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
13	12	rabex 5278	. . . . 5 ⊢ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ∈ V
14	9, 10, 11, 13	brpermmodelcnv 44978	. . . 4 ⊢ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ 𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
15		rabid 3416	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ↔ (𝑥 ∈ (𝐹‘𝑧) ∧ 𝜑))
16		vex 3440	. . . . . . 7 ⊢ 𝑧 ∈ V
17	9, 10, 11, 16	brpermmodel 44977	. . . . . 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 1795	. 2 ⊢ ∀𝑥(𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))
22	1, 8, 21	ceqsexv2d 3488	1 ⊢ ∃𝑦∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {crab 3394  Vcvv 3436   class class class wbr 5092   E cep 5518  ^◡ccnv 5618   ∘ ccom 5623  –1-1-onto→wf1o 6481  ‘cfv 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490

This theorem is referenced by: (None)