noinfcbv - Mathbox for Scott Fenton

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Theorem noinfcbv 33920

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), 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
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), 1_o⟩}), (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), 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))))
2		breq2 5078	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑦 <s 𝑥 ↔ 𝑦 <s 𝑎))
3	2	notbid 318	. . . . . 6 ⊢ (𝑥 = 𝑎 → (¬ 𝑦 <s 𝑥 ↔ ¬ 𝑦 <s 𝑎))
4	3	ralbidv 3112	. . . . 5 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑎))
5		breq1 5077	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑦 <s 𝑎 ↔ 𝑏 <s 𝑎))
6	5	notbid 318	. . . . . 6 ⊢ (𝑦 = 𝑏 → (¬ 𝑦 <s 𝑎 ↔ ¬ 𝑏 <s 𝑎))
7	6	cbvralvw 3383	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑎 ↔ ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎)
8	4, 7	bitrdi 287	. . . 4 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎))
9	8	cbvrexvw 3384	. . 3 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ∃𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎)
10	8	cbvriotavw 7242	. . . 4 ⊢ (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) = (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎)
11	10	dmeqi 5813	. . . . . 6 ⊢ dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) = dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎)
12	11	opeq1i 4807	. . . . 5 ⊢ ⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩ = ⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1_o⟩
13	12	sneqi 4572	. . . 4 ⊢ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩} = {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1_o⟩}
14	10, 13	uneq12i 4095	. . 3 ⊢ ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}) = ((℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1_o⟩})
15		eleq1w 2821	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → (𝑔 ∈ dom 𝑢 ↔ 𝑐 ∈ dom 𝑢))
16		suceq 6331	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑐 → suc 𝑔 = suc 𝑐)
17	16	reseq2d 5891	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑐 → (𝑢 ↾ suc 𝑔) = (𝑢 ↾ suc 𝑐))
18	16	reseq2d 5891	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑐 → (𝑣 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑐))
19	17, 18	eqeq12d 2754	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑐 → ((𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔) ↔ (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
20	19	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑔 = 𝑐 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
21	20	ralbidv 3112	. . . . . . . . 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 6783	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑐) = 𝑥))
23	15, 21, 22	3anbi123d 1435	. . . . . . . 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 3226	. . . . . . 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 6417	. . . . . 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 2750	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((𝑢‘𝑐) = 𝑥 ↔ (𝑢‘𝑐) = 𝑎))
27	26	3anbi3d 1441	. . . . . . . . 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 3226	. . . . . . . 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 5812	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑑 → dom 𝑢 = dom 𝑑)
30	29	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝑢 = 𝑑 → (𝑐 ∈ dom 𝑢 ↔ 𝑐 ∈ dom 𝑑))
31		breq1 5077	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑑 → (𝑢 <s 𝑣 ↔ 𝑑 <s 𝑣))
32	31	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑑 → (¬ 𝑢 <s 𝑣 ↔ ¬ 𝑑 <s 𝑣))
33		reseq1 5885	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑑 → (𝑢 ↾ suc 𝑐) = (𝑑 ↾ suc 𝑐))
34	33	eqeq1d 2740	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑑 → ((𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
35	32, 34	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑑 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
36	35	ralbidv 3112	. . . . . . . . . . 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 5078	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑒 → (𝑑 <s 𝑣 ↔ 𝑑 <s 𝑒))
38	37	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑒 → (¬ 𝑑 <s 𝑣 ↔ ¬ 𝑑 <s 𝑒))
39		reseq1 5885	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑒 → (𝑣 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))
40	39	eqeq2d 2749	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑒 → ((𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)))
41	38, 40	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑒 → ((¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))))
42	41	cbvralvw 3383	. . . . . . . . . . 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 287	. . . . . . . . . 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 6773	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑑 → (𝑢‘𝑐) = (𝑑‘𝑐))
45	44	eqeq1d 2740	. . . . . . . . . 10 ⊢ (𝑢 = 𝑑 → ((𝑢‘𝑐) = 𝑎 ↔ (𝑑‘𝑐) = 𝑎))
46	30, 43, 45	3anbi123d 1435	. . . . . . . . 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 3384	. . . . . . . 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 287	. . . . . . 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 6399	. . . . . 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 2794	. . . . 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 5187	. . . 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 2821	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 ∈ dom 𝑢 ↔ 𝑏 ∈ dom 𝑢))
53		suceq 6331	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
54	53	reseq2d 5891	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑏))
55	53	reseq2d 5891	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑏))
56	54, 55	eqeq12d 2754	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
57	56	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
58	57	ralbidv 3112	. . . . . . . . 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 3226	. . . . . . 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 2824	. . . . . . . . 9 ⊢ (𝑢 = 𝑑 → (𝑏 ∈ dom 𝑢 ↔ 𝑏 ∈ dom 𝑑))
62		reseq1 5885	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑑 → (𝑢 ↾ suc 𝑏) = (𝑑 ↾ suc 𝑏))
63	62	eqeq1d 2740	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑑 → ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) ↔ (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
64	32, 63	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑑 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ (¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
65	64	ralbidv 3112	. . . . . . . . . 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 5885	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑒 → (𝑣 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏))
67	66	eqeq2d 2749	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑒 → ((𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) ↔ (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))
68	38, 67	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑒 → ((¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏))))
69	68	cbvralvw 3383	. . . . . . . . . 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 287	. . . . . . . . 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 3384	. . . . . . 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 287	. . . . . 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 2811	. . . . 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 5170	. . . 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 2766	. . 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 4486	. 2 ⊢ 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)))) = if(∃𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎, ((℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1_o⟩}), (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 2766	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), 1_o⟩}), (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))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 {cab 2715 ∀wral 3064 ∃wrex 3065 ∪ cun 3885 ifcif 4459 {csn 4561 ⟨cop 4567 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ↾ cres 5591 suc csuc 6268 ℩cio 6389 ‘cfv 6433 ℩crio 7231 1_oc1o 8290 <s cslt 33844

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-xp 5595 df-dm 5599 df-res 5601 df-suc 6272 df-iota 6391 df-fv 6441 df-riota 7232

This theorem is referenced by: noeta 33946

W3C validator