Theorem noinfcbv 27065

Description: Change bound variables for surreal infimum. (Contributed by Scott Fenton, 9-Aug-2024.)

Hypothesis

Ref	Expression
noinfcbv.1	⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))

Assertion

Ref	Expression
noinfcbv	⊢ 𝑇 = if(∃𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎, ((℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎))))

Distinct variable groups:   𝑎,𝑏,𝑐,𝑑   𝑔,𝑎,𝑢,𝑣   𝑥,𝑎,𝑦,𝐵,𝑏,𝑐,𝑑   𝑒,𝑏,𝐵   𝐵,𝑔   𝑢,𝑏,𝐵,𝑣   𝑥,𝑏,𝐵,𝑦   𝑐,𝑑,𝑒,𝑔,𝑢,𝑣   𝑥,𝑐,𝑦   𝑒,𝑑,𝑔,𝑢,𝑣   𝑥,𝑑,𝑦   𝑒,𝑔,𝑢,𝑣   𝑥,𝑒,𝑦   𝑢,𝑔,𝑣   𝑥,𝑔,𝑦   𝑣,𝑢   𝑥,𝑢,𝑦   𝑦,𝑣   𝑥,𝑦

Allowed substitution hints:   𝑇(𝑥,𝑦,𝑣,𝑢,𝑒,𝑔,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem noinfcbv

Step	Hyp	Ref	Expression
1		noinfcbv.1	. 2 ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
2		breq2 5109	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑦 <s 𝑥 ↔ 𝑦 <s 𝑎))
3	2	notbid 317	. . . . . 6 ⊢ (𝑥 = 𝑎 → (¬ 𝑦 <s 𝑥 ↔ ¬ 𝑦 <s 𝑎))
4	3	ralbidv 3174	. . . . 5 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑎))
5		breq1 5108	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑦 <s 𝑎 ↔ 𝑏 <s 𝑎))
6	5	notbid 317	. . . . . 6 ⊢ (𝑦 = 𝑏 → (¬ 𝑦 <s 𝑎 ↔ ¬ 𝑏 <s 𝑎))
7	6	cbvralvw 3225	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑎 ↔ ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎)
8	4, 7	bitrdi 286	. . . 4 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎))
9	8	cbvrexvw 3226	. . 3 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ∃𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎)
10	8	cbvriotavw 7323	. . . 4 ⊢ (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) = (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎)
11	10	dmeqi 5860	. . . . . 6 ⊢ dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) = dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎)
12	11	opeq1i 4833	. . . . 5 ⊢ ⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩ = ⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1o⟩
13	12	sneqi 4597	. . . 4 ⊢ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩} = {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}
14	10, 13	uneq12i 4121	. . 3 ⊢ ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = ((℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1o⟩})
15		eleq1w 2820	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → (𝑔 ∈ dom 𝑢 ↔ 𝑐 ∈ dom 𝑢))
16		suceq 6383	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑐 → suc 𝑔 = suc 𝑐)
17	16	reseq2d 5937	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑐 → (𝑢 ↾ suc 𝑔) = (𝑢 ↾ suc 𝑐))
18	16	reseq2d 5937	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑐 → (𝑣 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑐))
19	17, 18	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑐 → ((𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔) ↔ (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
20	19	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑔 = 𝑐 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
21	20	ralbidv 3174	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
22		fveqeq2 6851	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑐) = 𝑥))
23	15, 21, 22	3anbi123d 1436	. . . . . . . 8 ⊢ (𝑔 = 𝑐 → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥)))
24	23	rexbidv 3175	. . . . . . 7 ⊢ (𝑔 = 𝑐 → (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ ∃𝑢 ∈ 𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥)))
25	24	iotabidv 6480	. . . . . 6 ⊢ (𝑔 = 𝑐 → (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = (℩𝑥∃𝑢 ∈ 𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥)))
26		eqeq2 2748	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((𝑢‘𝑐) = 𝑥 ↔ (𝑢‘𝑐) = 𝑎))
27	26	3anbi3d 1442	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥) ↔ (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑎)))
28	27	rexbidv 3175	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑢 ∈ 𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥) ↔ ∃𝑢 ∈ 𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑎)))
29		dmeq 5859	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑑 → dom 𝑢 = dom 𝑑)
30	29	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑢 = 𝑑 → (𝑐 ∈ dom 𝑢 ↔ 𝑐 ∈ dom 𝑑))
31		breq1 5108	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑑 → (𝑢 <s 𝑣 ↔ 𝑑 <s 𝑣))
32	31	notbid 317	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑑 → (¬ 𝑢 <s 𝑣 ↔ ¬ 𝑑 <s 𝑣))
33		reseq1 5931	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑑 → (𝑢 ↾ suc 𝑐) = (𝑑 ↾ suc 𝑐))
34	33	eqeq1d 2738	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑑 → ((𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
35	32, 34	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑑 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
36	35	ralbidv 3174	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑑 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
37		breq2 5109	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑒 → (𝑑 <s 𝑣 ↔ 𝑑 <s 𝑒))
38	37	notbid 317	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑒 → (¬ 𝑑 <s 𝑣 ↔ ¬ 𝑑 <s 𝑒))
39		reseq1 5931	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑒 → (𝑣 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))
40	39	eqeq2d 2747	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑒 → ((𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)))
41	38, 40	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑒 → ((¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))))
42	41	cbvralvw 3225	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ 𝐵 (¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)))
43	36, 42	bitrdi 286	. . . . . . . . . 10 ⊢ (𝑢 = 𝑑 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))))
44		fveq1 6841	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑑 → (𝑢‘𝑐) = (𝑑‘𝑐))
45	44	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑢 = 𝑑 → ((𝑢‘𝑐) = 𝑎 ↔ (𝑑‘𝑐) = 𝑎))
46	30, 43, 45	3anbi123d 1436	. . . . . . . . 9 ⊢ (𝑢 = 𝑑 → ((𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑎) ↔ (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎)))
47	46	cbvrexvw 3226	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑎) ↔ ∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎))
48	28, 47	bitrdi 286	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (∃𝑢 ∈ 𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥) ↔ ∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎)))
49	48	cbviotavw 6456	. . . . . 6 ⊢ (℩𝑥∃𝑢 ∈ 𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥)) = (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎))
50	25, 49	eqtrdi 2792	. . . . 5 ⊢ (𝑔 = 𝑐 → (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎)))
51	50	cbvmptv 5218	. . . 4 ⊢ (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = (𝑐 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎)))
52		eleq1w 2820	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 ∈ dom 𝑢 ↔ 𝑏 ∈ dom 𝑢))
53		suceq 6383	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
54	53	reseq2d 5937	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑏))
55	53	reseq2d 5937	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑏))
56	54, 55	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
57	56	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
58	57	ralbidv 3174	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
59	52, 58	anbi12d 631	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
60	59	rexbidv 3175	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
61	29	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑢 = 𝑑 → (𝑏 ∈ dom 𝑢 ↔ 𝑏 ∈ dom 𝑑))
62		reseq1 5931	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑑 → (𝑢 ↾ suc 𝑏) = (𝑑 ↾ suc 𝑏))
63	62	eqeq1d 2738	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑑 → ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) ↔ (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
64	32, 63	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑑 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ (¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
65	64	ralbidv 3174	. . . . . . . . . 10 ⊢ (𝑢 = 𝑑 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
66		reseq1 5931	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑒 → (𝑣 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏))
67	66	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑒 → ((𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) ↔ (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))
68	38, 67	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑒 → ((¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏))))
69	68	cbvralvw 3225	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ 𝐵 (¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))
70	65, 69	bitrdi 286	. . . . . . . . 9 ⊢ (𝑢 = 𝑑 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏))))
71	61, 70	anbi12d 631	. . . . . . . 8 ⊢ (𝑢 = 𝑑 → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) ↔ (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))))
72	71	cbvrexvw 3226	. . . . . . 7 ⊢ (∃𝑢 ∈ 𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) ↔ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏))))
73	60, 72	bitrdi 286	. . . . . 6 ⊢ (𝑦 = 𝑏 → (∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))))
74	73	cbvabv 2809	. . . . 5 ⊢ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} = {𝑏 ∣ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))}
75	74	mpteq1i 5201	. . . 4 ⊢ (𝑐 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎))) = (𝑐 ∈ {𝑏 ∣ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎)))
76	51, 75	eqtri 2764	. . 3 ⊢ (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = (𝑐 ∈ {𝑏 ∣ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎)))
77	9, 14, 76	ifbieq12i 4513	. 2 ⊢ if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) = if(∃𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎, ((℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎))))
78	1, 77	eqtri 2764	1 ⊢ 𝑇 = if(∃𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎, ((℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2713  ∀wral 3064  ∃wrex 3073   ∪ cun 3908  ifcif 4486  {csn 4586  ⟨cop 4592   class class class wbr 5105   ↦ cmpt 5188  dom cdm 5633   ↾ cres 5635  suc csuc 6319  ℩cio 6446  ‘cfv 6496  ℩crio 7312  1_oc1o 8405   <s cslt 26989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-xp 5639  df-dm 5643  df-res 5645  df-suc 6323  df-iota 6448  df-fv 6504  df-riota 7313

This theorem is referenced by:  noeta  27091