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Theorem noinfres 27705

Description: The restriction of surreal infimum when there is no minimum. (Contributed by Scott Fenton, 8-Aug-2024.)

**Hypothesis**
Ref	Expression
noinfres.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), 1_o⟩}), (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
noinfres	⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑇 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))

Distinct variable groups: 𝑢,𝐵,𝑣,𝑦,𝑔,𝑥 𝑔,𝑉 𝑣,𝐺 𝑢,𝑈,𝑣,𝑥 Allowed substitution hints: 𝑇(𝑥,𝑦,𝑣,𝑢,𝑔) 𝑈(𝑦,𝑔) 𝐺(𝑥,𝑦,𝑢,𝑔) 𝑉(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem noinfres Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmres 5979	. . . 4 ⊢ dom (𝑇 ↾ suc 𝐺) = (suc 𝐺 ∩ dom 𝑇)
2		noinfres.1	. . . . . . . . 9 ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (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))))
3	2	noinfno 27701	. . . . . . . 8 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
4	3	3ad2ant2 1135	. . . . . . 7 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝑇 ∈ No )
5		nodmord 27636	. . . . . . 7 ⊢ (𝑇 ∈ No → Ord dom 𝑇)
6	4, 5	syl 17	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → Ord dom 𝑇)
7		simp31 1211	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝑈 ∈ 𝐵)
8		simp32 1212	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝐺 ∈ dom 𝑈)
9		simp33 1213	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
10		dmeq 5860	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑈 → dom 𝑏 = dom 𝑈)
11	10	eleq2d 2823	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑈 → (𝐺 ∈ dom 𝑏 ↔ 𝐺 ∈ dom 𝑈))
12		breq1 5103	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑈 → (𝑏 <s 𝑐 ↔ 𝑈 <s 𝑐))
13	12	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑈 → (¬ 𝑏 <s 𝑐 ↔ ¬ 𝑈 <s 𝑐))
14		reseq1 5940	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑈 → (𝑏 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))
15	14	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑈 → ((𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺) ↔ (𝑈 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)))
16	13, 15	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑈 → ((¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)) ↔ (¬ 𝑈 <s 𝑐 → (𝑈 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺))))
17	16	ralbidv 3161	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑈 → (∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)) ↔ ∀𝑐 ∈ 𝐵 (¬ 𝑈 <s 𝑐 → (𝑈 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺))))
18		breq2 5104	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑣 → (𝑈 <s 𝑐 ↔ 𝑈 <s 𝑣))
19	18	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑣 → (¬ 𝑈 <s 𝑐 ↔ ¬ 𝑈 <s 𝑣))
20		reseq1 5940	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑣 → (𝑐 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))
21	20	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑣 → ((𝑈 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺) ↔ (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
22	19, 21	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑣 → ((¬ 𝑈 <s 𝑐 → (𝑈 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)) ↔ (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
23	22	cbvralvw 3216	. . . . . . . . . . . 12 ⊢ (∀𝑐 ∈ 𝐵 (¬ 𝑈 <s 𝑐 → (𝑈 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
24	17, 23	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑈 → (∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
25	11, 24	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑏 = 𝑈 → ((𝐺 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺))) ↔ (𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))))
26	25	rspcev 3578	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝐵 ∧ (𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → ∃𝑏 ∈ 𝐵 (𝐺 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺))))
27	7, 8, 9, 26	syl12anc 837	. . . . . . . 8 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → ∃𝑏 ∈ 𝐵 (𝐺 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺))))
28		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐺 → (𝑎 ∈ dom 𝑏 ↔ 𝐺 ∈ dom 𝑏))
29		suceq 6393	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐺 → suc 𝑎 = suc 𝐺)
30	29	reseq2d 5946	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐺 → (𝑏 ↾ suc 𝑎) = (𝑏 ↾ suc 𝐺))
31	29	reseq2d 5946	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐺 → (𝑐 ↾ suc 𝑎) = (𝑐 ↾ suc 𝐺))
32	30, 31	eqeq12d 2753	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐺 → ((𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎) ↔ (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)))
33	32	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐺 → ((¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎)) ↔ (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺))))
34	33	ralbidv 3161	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐺 → (∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎)) ↔ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺))))
35	28, 34	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐺 → ((𝑎 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎))) ↔ (𝐺 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)))))
36	35	rexbidv 3162	. . . . . . . . . 10 ⊢ (𝑎 = 𝐺 → (∃𝑏 ∈ 𝐵 (𝑎 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎))) ↔ ∃𝑏 ∈ 𝐵 (𝐺 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)))))
37	36	elabg 3633	. . . . . . . . 9 ⊢ (𝐺 ∈ dom 𝑈 → (𝐺 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐵 (𝑎 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎)))} ↔ ∃𝑏 ∈ 𝐵 (𝐺 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)))))
38	8, 37	syl 17	. . . . . . . 8 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝐺 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐵 (𝑎 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎)))} ↔ ∃𝑏 ∈ 𝐵 (𝐺 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)))))
39	27, 38	mpbird 257	. . . . . . 7 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝐺 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐵 (𝑎 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎)))})
40	2	noinfdm 27702	. . . . . . . 8 ⊢ (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = {𝑎 ∣ ∃𝑏 ∈ 𝐵 (𝑎 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎)))})
41	40	3ad2ant1 1134	. . . . . . 7 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → dom 𝑇 = {𝑎 ∣ ∃𝑏 ∈ 𝐵 (𝑎 ∈ dom 𝑏 ∧ ∀𝑐 ∈ 𝐵 (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎)))})
42	39, 41	eleqtrrd 2840	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝐺 ∈ dom 𝑇)
43		ordsucss 7770	. . . . . 6 ⊢ (Ord dom 𝑇 → (𝐺 ∈ dom 𝑇 → suc 𝐺 ⊆ dom 𝑇))
44	6, 42, 43	sylc 65	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → suc 𝐺 ⊆ dom 𝑇)
45		dfss2 3921	. . . . 5 ⊢ (suc 𝐺 ⊆ dom 𝑇 ↔ (suc 𝐺 ∩ dom 𝑇) = suc 𝐺)
46	44, 45	sylib 218	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (suc 𝐺 ∩ dom 𝑇) = suc 𝐺)
47	1, 46	eqtrid 2784	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → dom (𝑇 ↾ suc 𝐺) = suc 𝐺)
48		dmres 5979	. . . 4 ⊢ dom (𝑈 ↾ suc 𝐺) = (suc 𝐺 ∩ dom 𝑈)
49		simp2l 1201	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝐵 ⊆ No )
50	49, 7	sseldd 3936	. . . . . . . 8 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝑈 ∈ No )
51		nodmon 27633	. . . . . . . 8 ⊢ (𝑈 ∈ No → dom 𝑈 ∈ On)
52	50, 51	syl 17	. . . . . . 7 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → dom 𝑈 ∈ On)
53		eloni 6335	. . . . . . 7 ⊢ (dom 𝑈 ∈ On → Ord dom 𝑈)
54	52, 53	syl 17	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → Ord dom 𝑈)
55		ordsucss 7770	. . . . . 6 ⊢ (Ord dom 𝑈 → (𝐺 ∈ dom 𝑈 → suc 𝐺 ⊆ dom 𝑈))
56	54, 8, 55	sylc 65	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → suc 𝐺 ⊆ dom 𝑈)
57		dfss2 3921	. . . . 5 ⊢ (suc 𝐺 ⊆ dom 𝑈 ↔ (suc 𝐺 ∩ dom 𝑈) = suc 𝐺)
58	56, 57	sylib 218	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (suc 𝐺 ∩ dom 𝑈) = suc 𝐺)
59	48, 58	eqtrid 2784	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → dom (𝑈 ↾ suc 𝐺) = suc 𝐺)
60	47, 59	eqtr4d 2775	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → dom (𝑇 ↾ suc 𝐺) = dom (𝑈 ↾ suc 𝐺))
61	47	eleq2d 2823	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑎 ∈ dom (𝑇 ↾ suc 𝐺) ↔ 𝑎 ∈ suc 𝐺))
62		simpl1 1193	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
63		simpl2 1194	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉))
64		simpl31 1256	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → 𝑈 ∈ 𝐵)
65	56	sselda 3935	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → 𝑎 ∈ dom 𝑈)
66	50	adantr 480	. . . . . . . . . . . . 13 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → 𝑈 ∈ No )
67	66, 51	syl 17	. . . . . . . . . . . 12 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → dom 𝑈 ∈ On)
68		simpl32 1257	. . . . . . . . . . . 12 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → 𝐺 ∈ dom 𝑈)
69		onelon 6350	. . . . . . . . . . . 12 ⊢ ((dom 𝑈 ∈ On ∧ 𝐺 ∈ dom 𝑈) → 𝐺 ∈ On)
70	67, 68, 69	syl2anc 585	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → 𝐺 ∈ On)
71		onsucb 7769	. . . . . . . . . . 11 ⊢ (𝐺 ∈ On ↔ suc 𝐺 ∈ On)
72	70, 71	sylib 218	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → suc 𝐺 ∈ On)
73		eloni 6335	. . . . . . . . . 10 ⊢ (suc 𝐺 ∈ On → Ord suc 𝐺)
74	72, 73	syl 17	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → Ord suc 𝐺)
75		simpr 484	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → 𝑎 ∈ suc 𝐺)
76		ordsucss 7770	. . . . . . . . 9 ⊢ (Ord suc 𝐺 → (𝑎 ∈ suc 𝐺 → suc 𝑎 ⊆ suc 𝐺))
77	74, 75, 76	sylc 65	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → suc 𝑎 ⊆ suc 𝐺)
78		simpl33 1258	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
79		reseq1 5940	. . . . . . . . . . 11 ⊢ ((𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) → ((𝑈 ↾ suc 𝐺) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝐺) ↾ suc 𝑎))
80		resabs1 5973	. . . . . . . . . . . 12 ⊢ (suc 𝑎 ⊆ suc 𝐺 → ((𝑈 ↾ suc 𝐺) ↾ suc 𝑎) = (𝑈 ↾ suc 𝑎))
81		resabs1 5973	. . . . . . . . . . . 12 ⊢ (suc 𝑎 ⊆ suc 𝐺 → ((𝑣 ↾ suc 𝐺) ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))
82	80, 81	eqeq12d 2753	. . . . . . . . . . 11 ⊢ (suc 𝑎 ⊆ suc 𝐺 → (((𝑈 ↾ suc 𝐺) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝐺) ↾ suc 𝑎) ↔ (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
83	79, 82	imbitrid 244	. . . . . . . . . 10 ⊢ (suc 𝑎 ⊆ suc 𝐺 → ((𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
84	83	imim2d 57	. . . . . . . . 9 ⊢ (suc 𝑎 ⊆ suc 𝐺 → ((¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) → (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
85	84	ralimdv 3152	. . . . . . . 8 ⊢ (suc 𝑎 ⊆ suc 𝐺 → (∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) → ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
86	77, 78, 85	sylc 65	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
87	2	noinffv 27704	. . . . . . 7 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝑎 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))) → (𝑇‘𝑎) = (𝑈‘𝑎))
88	62, 63, 64, 65, 86, 87	syl113anc 1385	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → (𝑇‘𝑎) = (𝑈‘𝑎))
89	75	fvresd 6862	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ((𝑇 ↾ suc 𝐺)‘𝑎) = (𝑇‘𝑎))
90	75	fvresd 6862	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ((𝑈 ↾ suc 𝐺)‘𝑎) = (𝑈‘𝑎))
91	88, 89, 90	3eqtr4d 2782	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ((𝑇 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎))
92	91	ex 412	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑎 ∈ suc 𝐺 → ((𝑇 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎)))
93	61, 92	sylbid 240	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑎 ∈ dom (𝑇 ↾ suc 𝐺) → ((𝑇 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎)))
94	93	ralrimiv 3129	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → ∀𝑎 ∈ dom (𝑇 ↾ suc 𝐺)((𝑇 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎))
95		nofun 27632	. . . . 5 ⊢ (𝑇 ∈ No → Fun 𝑇)
96	95	funresd 6543	. . . 4 ⊢ (𝑇 ∈ No → Fun (𝑇 ↾ suc 𝐺))
97	4, 96	syl 17	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → Fun (𝑇 ↾ suc 𝐺))
98		nofun 27632	. . . . 5 ⊢ (𝑈 ∈ No → Fun 𝑈)
99	98	funresd 6543	. . . 4 ⊢ (𝑈 ∈ No → Fun (𝑈 ↾ suc 𝐺))
100	50, 99	syl 17	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → Fun (𝑈 ↾ suc 𝐺))
101		eqfunfv 6990	. . 3 ⊢ ((Fun (𝑇 ↾ suc 𝐺) ∧ Fun (𝑈 ↾ suc 𝐺)) → ((𝑇 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺) ↔ (dom (𝑇 ↾ suc 𝐺) = dom (𝑈 ↾ suc 𝐺) ∧ ∀𝑎 ∈ dom (𝑇 ↾ suc 𝐺)((𝑇 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎))))
102	97, 100, 101	syl2anc 585	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → ((𝑇 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺) ↔ (dom (𝑇 ↾ suc 𝐺) = dom (𝑈 ↾ suc 𝐺) ∧ ∀𝑎 ∈ dom (𝑇 ↾ suc 𝐺)((𝑇 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎))))
103	60, 94, 102	mpbir2and 714	1 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑇 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {cab 2715 ∀wral 3052 ∃wrex 3062 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ifcif 4481 {csn 4582 ⟨cop 4588 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5632 ↾ cres 5634 Ord word 6324 Oncon0 6325 suc csuc 6327 ℩cio 6454 Fun wfun 6494 ‘cfv 6500 ℩crio 7324 1_oc1o 8400 No csur 27622 <s clts 27623

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6328 df-on 6329 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-riota 7325 df-1o 8407 df-2o 8408 df-no 27625 df-lts 27626 df-bday 27627

This theorem is referenced by: noinfbnd1lem1 27706 noinfbnd2 27714

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