Theorem permaxrep 45748

Description: The Axiom of Replacement ax-rep 5240 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148.

Note that, to prove that an instance of Replacement 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 Replacement 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
permaxrep	⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))

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

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

Proof of Theorem permaxrep

Step	Hyp	Ref	Expression
1		nfa1 2189	. . . 4 ⊢ Ⅎ𝑦∀𝑦𝜑
2	1	mof 2593	. . 3 ⊢ (∃*𝑧∀𝑦𝜑 ↔ ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
3	2	albii 1852	. 2 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 ↔ ∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
4		fvex 6898	. . 3 ⊢ (◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ∈ V
5		nfmo1 2587	. . . . 5 ⊢ Ⅎ𝑧∃*𝑧∀𝑦𝜑
6	5	nfal 2358	. . . 4 ⊢ Ⅎ𝑧∀𝑤∃*𝑧∀𝑦𝜑
7		permmodel.1	. . . . . . 7 ⊢ 𝐹:V–1-1-onto→V
8		permmodel.2	. . . . . . 7 ⊢ 𝑅 = (◡𝐹 ∘ E )
9		vex 3461	. . . . . . 7 ⊢ 𝑧 ∈ V
10	7, 8, 9, 4	brpermmodel 45745	. . . . . 6 ⊢ (𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ 𝑧 ∈ (𝐹‘(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})))
11		fvex 6898	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
12		axrep6g 5253	. . . . . . . . 9 ⊢ (((𝐹‘𝑥) ∈ V ∧ ∀𝑤∃*𝑧∀𝑦𝜑) → {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑} ∈ V)
13	11, 12	mpan 703	. . . . . . . 8 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑} ∈ V)
14		f1ocnvfv2 7281	. . . . . . . 8 ⊢ ((𝐹:V–1-1-onto→V ∧ {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑} ∈ V) → (𝐹‘(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})) = {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
15	7, 13, 14	sylancr 599	. . . . . . 7 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → (𝐹‘(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})) = {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
16	15	eleq2d 2851	. . . . . 6 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → (𝑧 ∈ (𝐹‘(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})) ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}))
17	10, 16	bitrid 286	. . . . 5 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → (𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}))
18		df-rex 3092	. . . . . 6 ⊢ (∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑 ↔ ∃𝑤(𝑤 ∈ (𝐹‘𝑥) ∧ ∀𝑦𝜑))
19		abid 2747	. . . . . 6 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑} ↔ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑)
20		vex 3461	. . . . . . . . 9 ⊢ 𝑤 ∈ V
21		vex 3461	. . . . . . . . 9 ⊢ 𝑥 ∈ V
22	7, 8, 20, 21	brpermmodel 45745	. . . . . . . 8 ⊢ (𝑤𝑅𝑥 ↔ 𝑤 ∈ (𝐹‘𝑥))
23	22	anbi1i 636	. . . . . . 7 ⊢ ((𝑤𝑅𝑥 ∧ ∀𝑦𝜑) ↔ (𝑤 ∈ (𝐹‘𝑥) ∧ ∀𝑦𝜑))
24	23	exbii 1881	. . . . . 6 ⊢ (∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑) ↔ ∃𝑤(𝑤 ∈ (𝐹‘𝑥) ∧ ∀𝑦𝜑))
25	18, 19, 24	3bitr4i 306	. . . . 5 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑} ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑))
26	17, 25	bitrdi 290	. . . 4 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → (𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))
27	6, 26	alrimi 2252	. . 3 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → ∀𝑧(𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))
28		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑦◡𝐹
29		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑦(𝐹‘𝑥)
30	29, 1	nfrexw 3315	. . . . . 6 ⊢ Ⅎ𝑦∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑
31	30	nfab 2933	. . . . 5 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}
32	28, 31	nffv 6895	. . . 4 ⊢ Ⅎ𝑦(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
33		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑦𝑧
34		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑦𝑅
35	33, 34, 32	nfbr 5160	. . . . . 6 ⊢ Ⅎ𝑦 𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
36		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑤𝑅𝑥
37	36, 1	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑦(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)
38	37	nfex 2359	. . . . . 6 ⊢ Ⅎ𝑦∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)
39	35, 38	nfbi 1936	. . . . 5 ⊢ Ⅎ𝑦(𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑))
40	39	nfal 2358	. . . 4 ⊢ Ⅎ𝑦∀𝑧(𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑))
41		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑧◡𝐹
42		nfab1 2929	. . . . . . 7 ⊢ Ⅎ𝑧{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}
43	41, 42	nffv 6895	. . . . . 6 ⊢ Ⅎ𝑧(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
44	43	nfeq2 2944	. . . . 5 ⊢ Ⅎ𝑧 𝑦 = (◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
45		breq2 5115	. . . . . 6 ⊢ (𝑦 = (◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) → (𝑧𝑅𝑦 ↔ 𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})))
46	45	bibi1d 346	. . . . 5 ⊢ (𝑦 = (◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) → ((𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑))))
47	44, 46	albid 2261	. . . 4 ⊢ (𝑦 = (◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) → (∀𝑧(𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑))))
48	32, 40, 47	spcegf 3553	. . 3 ⊢ ((◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ∈ V → (∀𝑧(𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)) → ∃𝑦∀𝑧(𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑))))
49	4, 27, 48	mpsyl 69	. 2 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → ∃𝑦∀𝑧(𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))
50	3, 49	sylbir 238	1 ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃*wmo 2567  {cab 2743  ∃wrex 3091  Vcvv 3457   class class class wbr 5111   E cep 5562  ^◡ccnv 5662   ∘ ccom 5667  –1-1-onto→wf1o 6539  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548

This theorem is used by: (None)