Theorem permaxinf2lem 45608

Description: Lemma for permaxinf2 45609. (Contributed by Eric Schmidt, 6-Nov-2025.)

Hypotheses

Ref	Expression
permmodel.1	⊢ 𝐹:V–1-1-onto→V
permmodel.2	⊢ 𝑅 = (◡𝐹 ∘ E )
permaxinf2lem.3	⊢ 𝑍 = (rec((𝑣 ∈ V ↦ (◡𝐹‘((𝐹‘𝑣) ∪ {𝑣}))), (◡𝐹‘∅)) “ ω)

Assertion

Ref	Expression
permaxinf2lem	⊢ ∃𝑥(∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))

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

Allowed substitution hints:   𝑅(𝑥,𝑦,𝑤,𝑣)   𝑍(𝑤,𝑣)

Proof of Theorem permaxinf2lem

Step	Hyp	Ref	Expression
1		fvex 6892	. 2 ⊢ (◡𝐹‘𝑍) ∈ V
2		breq2 5114	. . . . 5 ⊢ (𝑥 = (◡𝐹‘𝑍) → (𝑦𝑅𝑥 ↔ 𝑦𝑅(◡𝐹‘𝑍)))
3	2	anbi1d 642	. . . 4 ⊢ (𝑥 = (◡𝐹‘𝑍) → ((𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ↔ (𝑦𝑅(◡𝐹‘𝑍) ∧ ∀𝑧 ¬ 𝑧𝑅𝑦)))
4	3	exbidv 1948	. . 3 ⊢ (𝑥 = (◡𝐹‘𝑍) → (∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦(𝑦𝑅(◡𝐹‘𝑍) ∧ ∀𝑧 ¬ 𝑧𝑅𝑦)))
5		breq2 5114	. . . . . . 7 ⊢ (𝑥 = (◡𝐹‘𝑍) → (𝑧𝑅𝑥 ↔ 𝑧𝑅(◡𝐹‘𝑍)))
6	5	anbi1d 642	. . . . . 6 ⊢ (𝑥 = (◡𝐹‘𝑍) → ((𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))) ↔ (𝑧𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))
7	6	exbidv 1948	. . . . 5 ⊢ (𝑥 = (◡𝐹‘𝑍) → (∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))) ↔ ∃𝑧(𝑧𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))
8	2, 7	imbi12d 347	. . . 4 ⊢ (𝑥 = (◡𝐹‘𝑍) → ((𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))) ↔ (𝑦𝑅(◡𝐹‘𝑍) → ∃𝑧(𝑧𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))))))
9	8	albidv 1947	. . 3 ⊢ (𝑥 = (◡𝐹‘𝑍) → (∀𝑦(𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))) ↔ ∀𝑦(𝑦𝑅(◡𝐹‘𝑍) → ∃𝑧(𝑧𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))))))
10	4, 9	anbi12d 643	. 2 ⊢ (𝑥 = (◡𝐹‘𝑍) → ((∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))))) ↔ (∃𝑦(𝑦𝑅(◡𝐹‘𝑍) ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅(◡𝐹‘𝑍) → ∃𝑧(𝑧𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))))
11		fvex 6892	. . . 4 ⊢ (◡𝐹‘∅) ∈ V
12		breq1 5113	. . . . 5 ⊢ (𝑦 = (◡𝐹‘∅) → (𝑦𝑅(◡𝐹‘𝑍) ↔ (◡𝐹‘∅)𝑅(◡𝐹‘𝑍)))
13		breq2 5114	. . . . . . 7 ⊢ (𝑦 = (◡𝐹‘∅) → (𝑧𝑅𝑦 ↔ 𝑧𝑅(◡𝐹‘∅)))
14	13	notbid 321	. . . . . 6 ⊢ (𝑦 = (◡𝐹‘∅) → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅(◡𝐹‘∅)))
15	14	albidv 1947	. . . . 5 ⊢ (𝑦 = (◡𝐹‘∅) → (∀𝑧 ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ¬ 𝑧𝑅(◡𝐹‘∅)))
16	12, 15	anbi12d 643	. . . 4 ⊢ (𝑦 = (◡𝐹‘∅) → ((𝑦𝑅(◡𝐹‘𝑍) ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ↔ ((◡𝐹‘∅)𝑅(◡𝐹‘𝑍) ∧ ∀𝑧 ¬ 𝑧𝑅(◡𝐹‘∅))))
17		orbitinit 45552	. . . . . . . 8 ⊢ ((◡𝐹‘∅) ∈ V → (◡𝐹‘∅) ∈ (rec((𝑣 ∈ V ↦ (◡𝐹‘((𝐹‘𝑣) ∪ {𝑣}))), (◡𝐹‘∅)) “ ω))
18		permaxinf2lem.3	. . . . . . . 8 ⊢ 𝑍 = (rec((𝑣 ∈ V ↦ (◡𝐹‘((𝐹‘𝑣) ∪ {𝑣}))), (◡𝐹‘∅)) “ ω)
19	17, 18	eleqtrrdi 2880	. . . . . . 7 ⊢ ((◡𝐹‘∅) ∈ V → (◡𝐹‘∅) ∈ 𝑍)
20	11, 19	ax-mp 5	. . . . . 6 ⊢ (◡𝐹‘∅) ∈ 𝑍
21		permmodel.1	. . . . . . 7 ⊢ 𝐹:V–1-1-onto→V
22		permmodel.2	. . . . . . 7 ⊢ 𝑅 = (◡𝐹 ∘ E )
23		orbitex 45551	. . . . . . . 8 ⊢ (rec((𝑣 ∈ V ↦ (◡𝐹‘((𝐹‘𝑣) ∪ {𝑣}))), (◡𝐹‘∅)) “ ω) ∈ V
24	18, 23	eqeltri 2865	. . . . . . 7 ⊢ 𝑍 ∈ V
25	21, 22, 11, 24	brpermmodelcnv 45600	. . . . . 6 ⊢ ((◡𝐹‘∅)𝑅(◡𝐹‘𝑍) ↔ (◡𝐹‘∅) ∈ 𝑍)
26	20, 25	mpbir 234	. . . . 5 ⊢ (◡𝐹‘∅)𝑅(◡𝐹‘𝑍)
27		noel 4299	. . . . . . 7 ⊢ ¬ 𝑧 ∈ ∅
28		vex 3467	. . . . . . . 8 ⊢ 𝑧 ∈ V
29		0ex 5269	. . . . . . . 8 ⊢ ∅ ∈ V
30	21, 22, 28, 29	brpermmodelcnv 45600	. . . . . . 7 ⊢ (𝑧𝑅(◡𝐹‘∅) ↔ 𝑧 ∈ ∅)
31	27, 30	mtbir 326	. . . . . 6 ⊢ ¬ 𝑧𝑅(◡𝐹‘∅)
32	31	ax-gen 1822	. . . . 5 ⊢ ∀𝑧 ¬ 𝑧𝑅(◡𝐹‘∅)
33	26, 32	pm3.2i 475	. . . 4 ⊢ ((◡𝐹‘∅)𝑅(◡𝐹‘𝑍) ∧ ∀𝑧 ¬ 𝑧𝑅(◡𝐹‘∅))
34	11, 16, 33	ceqsexv2d 3512	. . 3 ⊢ ∃𝑦(𝑦𝑅(◡𝐹‘𝑍) ∧ ∀𝑧 ¬ 𝑧𝑅𝑦)
35		fvex 6892	. . . . . . 7 ⊢ (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ∈ V
36		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑣𝑦
37		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑣(◡𝐹‘((𝐹‘𝑦) ∪ {𝑦}))
38		fveq2 6879	. . . . . . . . . 10 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
39		sneq 4601	. . . . . . . . . 10 ⊢ (𝑣 = 𝑦 → {𝑣} = {𝑦})
40	38, 39	uneq12d 4131	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → ((𝐹‘𝑣) ∪ {𝑣}) = ((𝐹‘𝑦) ∪ {𝑦}))
41	40	fveq2d 6883	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (◡𝐹‘((𝐹‘𝑣) ∪ {𝑣})) = (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})))
42	36, 37, 18, 41	orbitclmpt 45554	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑍 ∧ (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ∈ V) → (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ∈ 𝑍)
43	35, 42	mpan2 703	. . . . . 6 ⊢ (𝑦 ∈ 𝑍 → (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ∈ 𝑍)
44		vex 3467	. . . . . . 7 ⊢ 𝑦 ∈ V
45	21, 22, 44, 24	brpermmodelcnv 45600	. . . . . 6 ⊢ (𝑦𝑅(◡𝐹‘𝑍) ↔ 𝑦 ∈ 𝑍)
46	21, 22, 35, 24	brpermmodelcnv 45600	. . . . . 6 ⊢ ((◡𝐹‘((𝐹‘𝑦) ∪ {𝑦}))𝑅(◡𝐹‘𝑍) ↔ (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ∈ 𝑍)
47	43, 45, 46	3imtr4i 295	. . . . 5 ⊢ (𝑦𝑅(◡𝐹‘𝑍) → (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦}))𝑅(◡𝐹‘𝑍))
48		vex 3467	. . . . . . . 8 ⊢ 𝑤 ∈ V
49		fvex 6892	. . . . . . . . 9 ⊢ (𝐹‘𝑦) ∈ V
50		vsnex 5404	. . . . . . . . 9 ⊢ {𝑦} ∈ V
51	49, 50	unex 7739	. . . . . . . 8 ⊢ ((𝐹‘𝑦) ∪ {𝑦}) ∈ V
52	21, 22, 48, 51	brpermmodelcnv 45600	. . . . . . 7 ⊢ (𝑤𝑅(◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ↔ 𝑤 ∈ ((𝐹‘𝑦) ∪ {𝑦}))
53		elun 4115	. . . . . . 7 ⊢ (𝑤 ∈ ((𝐹‘𝑦) ∪ {𝑦}) ↔ (𝑤 ∈ (𝐹‘𝑦) ∨ 𝑤 ∈ {𝑦}))
54	21, 22, 48, 44	brpermmodel 45599	. . . . . . . . 9 ⊢ (𝑤𝑅𝑦 ↔ 𝑤 ∈ (𝐹‘𝑦))
55	54	bicomi 227	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐹‘𝑦) ↔ 𝑤𝑅𝑦)
56		velsn 4607	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦} ↔ 𝑤 = 𝑦)
57	55, 56	orbi12i 927	. . . . . . 7 ⊢ ((𝑤 ∈ (𝐹‘𝑦) ∨ 𝑤 ∈ {𝑦}) ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))
58	52, 53, 57	3bitri 300	. . . . . 6 ⊢ (𝑤𝑅(◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))
59	58	ax-gen 1822	. . . . 5 ⊢ ∀𝑤(𝑤𝑅(◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))
60		breq1 5113	. . . . . . 7 ⊢ (𝑧 = (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) → (𝑧𝑅(◡𝐹‘𝑍) ↔ (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦}))𝑅(◡𝐹‘𝑍)))
61		breq2 5114	. . . . . . . . 9 ⊢ (𝑧 = (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) → (𝑤𝑅𝑧 ↔ 𝑤𝑅(◡𝐹‘((𝐹‘𝑦) ∪ {𝑦}))))
62	61	bibi1d 346	. . . . . . . 8 ⊢ (𝑧 = (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) → ((𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)) ↔ (𝑤𝑅(◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))))
63	62	albidv 1947	. . . . . . 7 ⊢ (𝑧 = (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) → (∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)) ↔ ∀𝑤(𝑤𝑅(◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))))
64	60, 63	anbi12d 643	. . . . . 6 ⊢ (𝑧 = (◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) → ((𝑧𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))) ↔ ((◡𝐹‘((𝐹‘𝑦) ∪ {𝑦}))𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅(◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))
65	35, 64	spcev 3574	. . . . 5 ⊢ (((◡𝐹‘((𝐹‘𝑦) ∪ {𝑦}))𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅(◡𝐹‘((𝐹‘𝑦) ∪ {𝑦})) ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))) → ∃𝑧(𝑧𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))))
66	47, 59, 65	sylancl 597	. . . 4 ⊢ (𝑦𝑅(◡𝐹‘𝑍) → ∃𝑧(𝑧𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))))
67	66	ax-gen 1822	. . 3 ⊢ ∀𝑦(𝑦𝑅(◡𝐹‘𝑍) → ∃𝑧(𝑧𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦))))
68	34, 67	pm3.2i 475	. 2 ⊢ (∃𝑦(𝑦𝑅(◡𝐹‘𝑍) ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅(◡𝐹‘𝑍) → ∃𝑧(𝑧𝑅(◡𝐹‘𝑍) ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))
69	1, 10, 68	ceqsexv2d 3512	1 ⊢ ∃𝑥(∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463   ∪ cun 3911  ∅c0 4294  {csn 4591   class class class wbr 5110   ↦ cmpt 5193   E cep 5558  ^◡ccnv 5658   “ cima 5662   ∘ ccom 5663  –1-1-onto→wf1o 6533  ‘cfv 6534  ωcom 7858  reccrdg 8392

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-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  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-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393

This theorem is referenced by:  permaxinf2  45609