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Theorem noinfres 27886

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 6011	. . . 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 27882	. . . . . . . 8 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
4	3	3ad2ant2 1152	. . . . . . 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 27817	. . . . . . 7 ⊢ (𝑇 ∈ No → Ord dom 𝑇)
6	4, 5	syl 18	. . . . . 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 1228	. . . . . . . . 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 1229	. . . . . . . . 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 1230	. . . . . . . . 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 5893	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑈 → dom 𝑏 = dom 𝑈)
11	10	eleq2d 2849	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑈 → (𝐺 ∈ dom 𝑏 ↔ 𝐺 ∈ dom 𝑈))
12		breq1 5112	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑈 → (𝑏 <s 𝑐 ↔ 𝑈 <s 𝑐))
13	12	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑈 → (¬ 𝑏 <s 𝑐 ↔ ¬ 𝑈 <s 𝑐))
14		reseq1 5972	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑈 → (𝑏 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))
15	14	eqeq1d 2765	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑈 → ((𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺) ↔ (𝑈 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)))
16	13, 15	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑈 → ((¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)) ↔ (¬ 𝑈 <s 𝑐 → (𝑈 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺))))
17	16	ralbidv 3188	. . . . . . . . . . . 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 5113	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑣 → (𝑈 <s 𝑐 ↔ 𝑈 <s 𝑣))
19	18	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑣 → (¬ 𝑈 <s 𝑐 ↔ ¬ 𝑈 <s 𝑣))
20		reseq1 5972	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑣 → (𝑐 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))
21	20	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑣 → ((𝑈 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺) ↔ (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
22	19, 21	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑣 → ((¬ 𝑈 <s 𝑐 → (𝑈 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)) ↔ (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
23	22	cbvralvw 3243	. . . . . . . . . . . 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 290	. . . . . . . . . . 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 643	. . . . . . . . . 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 3581	. . . . . . . . 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 849	. . . . . . . 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 2851	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐺 → (𝑎 ∈ dom 𝑏 ↔ 𝐺 ∈ dom 𝑏))
29		suceq 6429	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐺 → suc 𝑎 = suc 𝐺)
30	29	reseq2d 5978	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐺 → (𝑏 ↾ suc 𝑎) = (𝑏 ↾ suc 𝐺))
31	29	reseq2d 5978	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐺 → (𝑐 ↾ suc 𝑎) = (𝑐 ↾ suc 𝐺))
32	30, 31	eqeq12d 2779	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐺 → ((𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎) ↔ (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺)))
33	32	imbi2d 343	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐺 → ((¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝑎) = (𝑐 ↾ suc 𝑎)) ↔ (¬ 𝑏 <s 𝑐 → (𝑏 ↾ suc 𝐺) = (𝑐 ↾ suc 𝐺))))
34	33	ralbidv 3188	. . . . . . . . . . . 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 643	. . . . . . . . . . 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 3189	. . . . . . . . . 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 3635	. . . . . . . . 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 18	. . . . . . . 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 260	. . . . . . 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 27883	. . . . . . . 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 1151	. . . . . . 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 2866	. . . . . 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 7810	. . . . . 6 ⊢ (Ord dom 𝑇 → (𝐺 ∈ dom 𝑇 → suc 𝐺 ⊆ dom 𝑇))
44	6, 42, 43	sylc 66	. . . . 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 3923	. . . . 5 ⊢ (suc 𝐺 ⊆ dom 𝑇 ↔ (suc 𝐺 ∩ dom 𝑇) = suc 𝐺)
46	44, 45	sylib 221	. . . 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 2810	. . 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 6011	. . . 4 ⊢ dom (𝑈 ↾ suc 𝐺) = (suc 𝐺 ∩ dom 𝑈)
49		simp2l 1218	. . . . . . . . 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 3938	. . . . . . . 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 27814	. . . . . . . 8 ⊢ (𝑈 ∈ No → dom 𝑈 ∈ On)
52	50, 51	syl 18	. . . . . . 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 6370	. . . . . . 7 ⊢ (dom 𝑈 ∈ On → Ord dom 𝑈)
54	52, 53	syl 18	. . . . . 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 7810	. . . . . 6 ⊢ (Ord dom 𝑈 → (𝐺 ∈ dom 𝑈 → suc 𝐺 ⊆ dom 𝑈))
56	54, 8, 55	sylc 66	. . . . 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 3923	. . . . 5 ⊢ (suc 𝐺 ⊆ dom 𝑈 ↔ (suc 𝐺 ∩ dom 𝑈) = suc 𝐺)
58	56, 57	sylib 221	. . . 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 2810	. . 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 2801	. 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 2849	. . . 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 1210	. . . . . . 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 1211	. . . . . . 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 1273	. . . . . . 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 3937	. . . . . . 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 485	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 6385	. . . . . . . . . . . 12 ⊢ ((dom 𝑈 ∈ On ∧ 𝐺 ∈ dom 𝑈) → 𝐺 ∈ On)
70	67, 68, 69	syl2anc 595	. . . . . . . . . . 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 7809	. . . . . . . . . . 11 ⊢ (𝐺 ∈ On ↔ suc 𝐺 ∈ On)
72	70, 71	sylib 221	. . . . . . . . . 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 6370	. . . . . . . . . 10 ⊢ (suc 𝐺 ∈ On → Ord suc 𝐺)
74	72, 73	syl 18	. . . . . . . . 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 489	. . . . . . . . 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 7810	. . . . . . . . 9 ⊢ (Ord suc 𝐺 → (𝑎 ∈ suc 𝐺 → suc 𝑎 ⊆ suc 𝐺))
77	74, 75, 76	sylc 66	. . . . . . . 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 1275	. . . . . . . 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 5972	. . . . . . . . . . 11 ⊢ ((𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) → ((𝑈 ↾ suc 𝐺) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝐺) ↾ suc 𝑎))
80		resabs1 6005	. . . . . . . . . . . 12 ⊢ (suc 𝑎 ⊆ suc 𝐺 → ((𝑈 ↾ suc 𝐺) ↾ suc 𝑎) = (𝑈 ↾ suc 𝑎))
81		resabs1 6005	. . . . . . . . . . . 12 ⊢ (suc 𝑎 ⊆ suc 𝐺 → ((𝑣 ↾ suc 𝐺) ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))
82	80, 81	eqeq12d 2779	. . . . . . . . . . 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 247	. . . . . . . . . 10 ⊢ (suc 𝑎 ⊆ suc 𝐺 → ((𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
84	83	imim2d 58	. . . . . . . . 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 3179	. . . . . . . 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 66	. . . . . . 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 27885	. . . . . . 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 1409	. . . . . 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 6901	. . . . . 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 6901	. . . . . 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 2808	. . . . 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 417	. . . 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 243	. . 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 3156	. 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 27813	. . . . 5 ⊢ (𝑇 ∈ No → Fun 𝑇)
96	95	funresd 6579	. . . 4 ⊢ (𝑇 ∈ No → Fun (𝑇 ↾ suc 𝐺))
97	4, 96	syl 18	. . 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 27813	. . . . 5 ⊢ (𝑈 ∈ No → Fun 𝑈)
99	98	funresd 6579	. . . 4 ⊢ (𝑈 ∈ No → Fun (𝑈 ↾ suc 𝐺))
100	50, 99	syl 18	. . 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 7031	. . 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 595	. 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 725	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 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 {cab 2741 ∀wral 3079 ∃wrex 3089 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ifcif 4487 {csn 4589 ⟨cop 4595 class class class wbr 5109 ↦ cmpt 5192 dom cdm 5661 ↾ cres 5663 Ord word 6359 Oncon0 6360 suc csuc 6362 ℩cio 6490 Fun wfun 6530 ‘cfv 6536 ℩crio 7366 1_oc1o 8442 No csur 27804 <s clts 27805

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-ord 6363 df-on 6364 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-riota 7367 df-1o 8449 df-2o 8450 df-no 27807 df-lts 27808 df-bday 27809

This theorem is referenced by: noinfbnd1lem1 27887 noinfbnd2 27895

W3C validator