Theorem permaxsep 45544

Description: The Axiom of Separation ax-sep 5243 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 6875	. 2 ⊢ (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ∈ V
2		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑥◡𝐹
3		nfrab1 3433	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}
4	2, 3	nffv 6872	. . . 4 ⊢ Ⅎ𝑥(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
5	4	nfeq2 2940	. . 3 ⊢ Ⅎ𝑥 𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
6		breq2 5101	. . . 4 ⊢ (𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) → (𝑥𝑅𝑦 ↔ 𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})))
7	6	bibi1d 345	. . 3 ⊢ (𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) → ((𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑)) ↔ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))))
8	5, 7	albid 2256	. 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 3457	. . . . 5 ⊢ 𝑥 ∈ V
12		fvex 6875	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
13	12	rabex 5292	. . . . 5 ⊢ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ∈ V
14	9, 10, 11, 13	brpermmodelcnv 45541	. . . 4 ⊢ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ 𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑})
15		rabid 3434	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ↔ (𝑥 ∈ (𝐹‘𝑧) ∧ 𝜑))
16		vex 3457	. . . . . . 7 ⊢ 𝑧 ∈ V
17	9, 10, 11, 16	brpermmodel 45540	. . . . . 6 ⊢ (𝑥𝑅𝑧 ↔ 𝑥 ∈ (𝐹‘𝑧))
18	17	bicomi 226	. . . . 5 ⊢ (𝑥 ∈ (𝐹‘𝑧) ↔ 𝑥𝑅𝑧)
19	15, 18	bianbi 636	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ↔ (𝑥𝑅𝑧 ∧ 𝜑))
20	14, 19	bitri 277	. . 3 ⊢ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))
21	20	ax-gen 1814	. 2 ⊢ ∀𝑥(𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑))
22	1, 8, 21	ceqsexv2d 3502	1 ⊢ ∃𝑦∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {crab 3413  Vcvv 3453   class class class wbr 5097   E cep 5542  ^◡ccnv 5642   ∘ ccom 5647  –1-1-onto→wf1o 6515  ‘cfv 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-eprel 5543  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524

This theorem is referenced by: (None)