Theorem permaxrep 45163

Description: The Axiom of Replacement ax-rep 5221 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 2156	. . . 4 ⊢ Ⅎ𝑦∀𝑦𝜑
2	1	mof 2560	. . 3 ⊢ (∃*𝑧∀𝑦𝜑 ↔ ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
3	2	albii 1820	. 2 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 ↔ ∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
4		fvex 6844	. . 3 ⊢ (◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ∈ V
5		nfmo1 2554	. . . . 5 ⊢ Ⅎ𝑧∃*𝑧∀𝑦𝜑
6	5	nfal 2326	. . . 4 ⊢ Ⅎ𝑧∀𝑤∃*𝑧∀𝑦𝜑
7		permmodel.1	. . . . . . 7 ⊢ 𝐹:V–1-1-onto→V
8		permmodel.2	. . . . . . 7 ⊢ 𝑅 = (◡𝐹 ∘ E )
9		vex 3441	. . . . . . 7 ⊢ 𝑧 ∈ V
10	7, 8, 9, 4	brpermmodel 45160	. . . . . 6 ⊢ (𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ 𝑧 ∈ (𝐹‘(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})))
11		fvex 6844	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
12		axrep6g 5232	. . . . . . . . 9 ⊢ (((𝐹‘𝑥) ∈ V ∧ ∀𝑤∃*𝑧∀𝑦𝜑) → {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑} ∈ V)
13	11, 12	mpan 690	. . . . . . . 8 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑} ∈ V)
14		f1ocnvfv2 7220	. . . . . . . 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 587	. . . . . . 7 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → (𝐹‘(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})) = {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
16	15	eleq2d 2819	. . . . . 6 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → (𝑧 ∈ (𝐹‘(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})) ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}))
17	10, 16	bitrid 283	. . . . 5 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → (𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}))
18		df-rex 3058	. . . . . 6 ⊢ (∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑 ↔ ∃𝑤(𝑤 ∈ (𝐹‘𝑥) ∧ ∀𝑦𝜑))
19		abid 2715	. . . . . 6 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑} ↔ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑)
20		vex 3441	. . . . . . . . 9 ⊢ 𝑤 ∈ V
21		vex 3441	. . . . . . . . 9 ⊢ 𝑥 ∈ V
22	7, 8, 20, 21	brpermmodel 45160	. . . . . . . 8 ⊢ (𝑤𝑅𝑥 ↔ 𝑤 ∈ (𝐹‘𝑥))
23	22	anbi1i 624	. . . . . . 7 ⊢ ((𝑤𝑅𝑥 ∧ ∀𝑦𝜑) ↔ (𝑤 ∈ (𝐹‘𝑥) ∧ ∀𝑦𝜑))
24	23	exbii 1849	. . . . . 6 ⊢ (∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑) ↔ ∃𝑤(𝑤 ∈ (𝐹‘𝑥) ∧ ∀𝑦𝜑))
25	18, 19, 24	3bitr4i 303	. . . . 5 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑} ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑))
26	17, 25	bitrdi 287	. . . 4 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → (𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))
27	6, 26	alrimi 2218	. . 3 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → ∀𝑧(𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))
28		nfcv 2895	. . . . 5 ⊢ Ⅎ𝑦◡𝐹
29		nfcv 2895	. . . . . . 7 ⊢ Ⅎ𝑦(𝐹‘𝑥)
30	29, 1	nfrexw 3281	. . . . . 6 ⊢ Ⅎ𝑦∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑
31	30	nfab 2901	. . . . 5 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}
32	28, 31	nffv 6841	. . . 4 ⊢ Ⅎ𝑦(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
33		nfcv 2895	. . . . . . 7 ⊢ Ⅎ𝑦𝑧
34		nfcv 2895	. . . . . . 7 ⊢ Ⅎ𝑦𝑅
35	33, 34, 32	nfbr 5142	. . . . . 6 ⊢ Ⅎ𝑦 𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
36		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑤𝑅𝑥
37	36, 1	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑦(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)
38	37	nfex 2327	. . . . . 6 ⊢ Ⅎ𝑦∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)
39	35, 38	nfbi 1904	. . . . 5 ⊢ Ⅎ𝑦(𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑))
40	39	nfal 2326	. . . 4 ⊢ Ⅎ𝑦∀𝑧(𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑))
41		nfcv 2895	. . . . . . 7 ⊢ Ⅎ𝑧◡𝐹
42		nfab1 2897	. . . . . . 7 ⊢ Ⅎ𝑧{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}
43	41, 42	nffv 6841	. . . . . 6 ⊢ Ⅎ𝑧(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
44	43	nfeq2 2913	. . . . 5 ⊢ Ⅎ𝑧 𝑦 = (◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})
45		breq2 5099	. . . . . 6 ⊢ (𝑦 = (◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑}) → (𝑧𝑅𝑦 ↔ 𝑧𝑅(◡𝐹‘{𝑧 ∣ ∃𝑤 ∈ (𝐹‘𝑥)∀𝑦𝜑})))
46	45	bibi1d 343	. . . . 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 2227	. . . 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 3543	. . 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 68	. 2 ⊢ (∀𝑤∃*𝑧∀𝑦𝜑 → ∃𝑦∀𝑧(𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))
50	3, 49	sylbir 235	1 ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃*wmo 2535  {cab 2711  ∃wrex 3057  Vcvv 3437   class class class wbr 5095   E cep 5520  ^◡ccnv 5620   ∘ ccom 5625  –1-1-onto→wf1o 6488  ‘cfv 6489

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 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497

This theorem is referenced by: (None)