Theorem noinfdm 27785

Description: Next, we calculate the domain of 𝑇. This is mostly to change bound variables. (Contributed by Scott Fenton, 8-Aug-2024.)

Hypothesis

Ref	Expression
noinfdm.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
noinfdm	⊢ (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})

Distinct variable groups:   𝐵,𝑔   𝐵,𝑝,𝑞,𝑢,𝑣,𝑦,𝑧   𝑢,𝑔,𝑣,𝑦

Allowed substitution hints:   𝐵(𝑥)   𝑇(𝑥,𝑦,𝑧,𝑣,𝑢,𝑔,𝑞,𝑝)

Proof of Theorem noinfdm

Step	Hyp	Ref	Expression
1		noinfdm.1	. . . 4 ⊢ 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		iffalse 4491	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → 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)))) = (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	1, 2	eqtrid 2811	. . 3 ⊢ (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → 𝑇 = (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))))
4	3	dmeqd 5883	. 2 ⊢ (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = dom (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))))
5		iotaex 6499	. . . 4 ⊢ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) ∈ V
6		eqid 2764	. . . 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))) = (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)))
7	5, 6	dmmpti 6667	. . 3 ⊢ dom (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))) = {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
8		eleq1w 2847	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ dom 𝑢 ↔ 𝑧 ∈ dom 𝑢))
9		suceq 6416	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
10	9	reseq2d 5967	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑧))
11	9	reseq2d 5967	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑧))
12	10, 11	eqeq12d 2780	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)))
13	12	imbi2d 342	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))))
14	13	ralbidv 3187	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))))
15	8, 14	anbi12d 641	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑧 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)))))
16	15	rexbidv 3188	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐵 (𝑧 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)))))
17		dmeq 5881	. . . . . . . 8 ⊢ (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
18	17	eleq2d 2850	. . . . . . 7 ⊢ (𝑢 = 𝑝 → (𝑧 ∈ dom 𝑢 ↔ 𝑧 ∈ dom 𝑝))
19		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑝 → (𝑢 <s 𝑣 ↔ 𝑝 <s 𝑣))
20	19	notbid 320	. . . . . . . . . 10 ⊢ (𝑢 = 𝑝 → (¬ 𝑢 <s 𝑣 ↔ ¬ 𝑝 <s 𝑣))
21		reseq1 5961	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑝 → (𝑢 ↾ suc 𝑧) = (𝑝 ↾ suc 𝑧))
22	21	eqeq1d 2766	. . . . . . . . . 10 ⊢ (𝑢 = 𝑝 → ((𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧) ↔ (𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)))
23	20, 22	imbi12d 346	. . . . . . . . 9 ⊢ (𝑢 = 𝑝 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)) ↔ (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))))
24	23	ralbidv 3187	. . . . . . . 8 ⊢ (𝑢 = 𝑝 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))))
25		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑞 → (𝑝 <s 𝑣 ↔ 𝑝 <s 𝑞))
26	25	notbid 320	. . . . . . . . . 10 ⊢ (𝑣 = 𝑞 → (¬ 𝑝 <s 𝑣 ↔ ¬ 𝑝 <s 𝑞))
27		reseq1 5961	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑞 → (𝑣 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))
28	27	eqeq2d 2775	. . . . . . . . . 10 ⊢ (𝑣 = 𝑞 → ((𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧) ↔ (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))
29	26, 28	imbi12d 346	. . . . . . . . 9 ⊢ (𝑣 = 𝑞 → ((¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)) ↔ (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))))
30	29	cbvralvw 3242	. . . . . . . 8 ⊢ (∀𝑣 ∈ 𝐵 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)) ↔ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))
31	24, 30	bitrdi 289	. . . . . . 7 ⊢ (𝑢 = 𝑝 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)) ↔ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))))
32	18, 31	anbi12d 641	. . . . . 6 ⊢ (𝑢 = 𝑝 → ((𝑧 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))) ↔ (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))))
33	32	cbvrexvw 3243	. . . . 5 ⊢ (∃𝑢 ∈ 𝐵 (𝑧 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))) ↔ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))))
34	16, 33	bitrdi 289	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))))
35	34	cbvabv 2834	. . 3 ⊢ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} = {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}
36	7, 35	eqtri 2787	. 2 ⊢ dom (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))) = {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}
37	4, 36	eqtrdi 2815	1 ⊢ (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {cab 2742  ∀wral 3078  ∃wrex 3088   ∪ cun 3904  ifcif 4482  {csn 4584  ⟨cop 4590   class class class wbr 5102   ↦ cmpt 5183  dom cdm 5649   ↾ cres 5651  suc csuc 6350  ℩cio 6477  ‘cfv 6523  ℩crio 7354  1_oc1o 8432   <s clts 27707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-res 5661  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526

This theorem is referenced by:  noinfbday  27786  noinfres  27788  noinfbnd1lem1  27789  noinfbnd1lem3  27791  noinfbnd1lem5  27793  noinfbnd2  27797