Theorem permaxsep 45607

Description: The Axiom of Separation ax-sep 5261 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 6895	. 2 ⊢ (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ∈ V
2		nfcv 2931	. . . . 5 ⊢ Ⅎ𝑥◡𝐹
3		nfrab1 3443	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}
4	2, 3	nffv 6892	. . . 4 ⊢ Ⅎ𝑥(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
5	4	nfeq2 2948	. . 3 ⊢ Ⅎ𝑥 𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
6		breq2 5117	. . . 4 ⊢ (𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) → (𝑥𝑅𝑦 ↔ 𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})))
7	6	bibi1d 346	. . 3 ⊢ (𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) → ((𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑)) ↔ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))))
8	5, 7	albid 2264	. 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 3467	. . . . 5 ⊢ 𝑥 ∈ V
12		fvex 6895	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
13	12	rabex 5310	. . . . 5 ⊢ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ∈ V
14	9, 10, 11, 13	brpermmodelcnv 45604	. . . 4 ⊢ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ 𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
15		rabid 3444	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ↔ (𝑥 ∈ (𝐹‘𝑧) ∧ 𝜑))
16		vex 3467	. . . . . . 7 ⊢ 𝑧 ∈ V
17	9, 10, 11, 16	brpermmodel 45603	. . . . . 6 ⊢ (𝑥𝑅𝑧 ↔ 𝑥 ∈ (𝐹‘𝑧))
18	17	bicomi 227	. . . . 5 ⊢ (𝑥 ∈ (𝐹‘𝑧) ↔ 𝑥𝑅𝑧)
19	15, 18	bianbi 638	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ↔ (𝑥𝑅𝑧 ∧ 𝜑))
20	14, 19	bitri 278	. . 3 ⊢ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))
21	20	ax-gen 1822	. 2 ⊢ ∀𝑥(𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))
22	1, 8, 21	ceqsexv2d 3512	1 ⊢ ∃𝑦∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {crab 3423  Vcvv 3463   class class class wbr 5113   E cep 5561  ^◡ccnv 5661   ∘ ccom 5666  –1-1-onto→wf1o 6536  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by: (None)