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Theorem nosupcbv 27875

Description: Lemma to change bound variables in a surreal supremum. (Contributed by Scott Fenton, 9-Aug-2024.)

**Hypothesis**
Ref	Expression
nosupcbv.1	⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))

**Assertion**
Ref	Expression
nosupcbv	⊢ 𝑆 = if(∃𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏, ((℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2_o⟩}), (𝑐 ∈ {𝑑 ∣ ∃𝑒 ∈ 𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎))))

Distinct variable groups: 𝐴,𝑎,𝑐,𝑒,𝑔,𝑢,𝑣,𝑦 𝐴,𝑑,𝑓,𝑐,𝑒,𝑢,𝑣,𝑦 𝐴,𝑏,𝑥,𝑎,𝑦 𝑥,𝑐,𝑒 𝑓,𝑔,𝑥,𝑢 Allowed substitution hints: 𝑆(𝑥, 𝑦, 𝑣, 𝑢, 𝑒, 𝑓, 𝑔, 𝑎, 𝑏, 𝑐, 𝑑)

**Proof of Theorem nosupcbv**
Step	Hyp	Ref	Expression
1		nosupcbv.1	. 2 ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
2		breq1 5112	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 <s 𝑦 ↔ 𝑎 <s 𝑦))
3	2	notbid 321	. . . . . 6 ⊢ (𝑥 = 𝑎 → (¬ 𝑥 <s 𝑦 ↔ ¬ 𝑎 <s 𝑦))
4	3	ralbidv 3188	. . . . 5 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑎 <s 𝑦))
5		breq2 5113	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑎 <s 𝑦 ↔ 𝑎 <s 𝑏))
6	5	notbid 321	. . . . . 6 ⊢ (𝑦 = 𝑏 → (¬ 𝑎 <s 𝑦 ↔ ¬ 𝑎 <s 𝑏))
7	6	cbvralvw 3243	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑎 <s 𝑦 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏)
8	4, 7	bitrdi 290	. . . 4 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏))
9	8	cbvrexvw 3244	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ ∃𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏)
10	8	cbvriotavw 7377	. . . 4 ⊢ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏)
11	10	dmeqi 5894	. . . . . 6 ⊢ dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏)
12	11	opeq1i 4841	. . . . 5 ⊢ ⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩ = ⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2_o⟩
13	12	sneqi 4600	. . . 4 ⊢ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩} = {⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2_o⟩}
14	10, 13	uneq12i 4120	. . 3 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) = ((℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2_o⟩})
15		eleq1w 2846	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → (𝑔 ∈ dom 𝑢 ↔ 𝑐 ∈ dom 𝑢))
16		suceq 6429	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑐 → suc 𝑔 = suc 𝑐)
17	16	reseq2d 5978	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑐 → (𝑢 ↾ suc 𝑔) = (𝑢 ↾ suc 𝑐))
18	16	reseq2d 5978	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑐 → (𝑣 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑐))
19	17, 18	eqeq12d 2779	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑐 → ((𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔) ↔ (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
20	19	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑔 = 𝑐 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
21	20	ralbidv 3188	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
22		fveqeq2 6890	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑐) = 𝑥))
23	15, 21, 22	3anbi123d 1464	. . . . . . . 8 ⊢ (𝑔 = 𝑐 → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥)))
24	23	rexbidv 3189	. . . . . . 7 ⊢ (𝑔 = 𝑐 → (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ ∃𝑢 ∈ 𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥)))
25	24	iotabidv 6520	. . . . . 6 ⊢ (𝑔 = 𝑐 → (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = (℩𝑥∃𝑢 ∈ 𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥)))
26		eqeq2 2775	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((𝑢‘𝑐) = 𝑥 ↔ (𝑢‘𝑐) = 𝑎))
27	26	3anbi3d 1470	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥) ↔ (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑎)))
28	27	rexbidv 3189	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑢 ∈ 𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥) ↔ ∃𝑢 ∈ 𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑎)))
29		dmeq 5893	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑒 → dom 𝑢 = dom 𝑒)
30	29	eleq2d 2849	. . . . . . . . . 10 ⊢ (𝑢 = 𝑒 → (𝑐 ∈ dom 𝑢 ↔ 𝑐 ∈ dom 𝑒))
31		breq2 5113	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑒 → (𝑣 <s 𝑢 ↔ 𝑣 <s 𝑒))
32	31	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑒 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑒))
33		reseq1 5972	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑒 → (𝑢 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))
34	33	eqeq1d 2765	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑒 → ((𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
35	32, 34	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑒 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
36	35	ralbidv 3188	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑒 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
37		breq1 5112	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑓 → (𝑣 <s 𝑒 ↔ 𝑓 <s 𝑒))
38	37	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (¬ 𝑣 <s 𝑒 ↔ ¬ 𝑓 <s 𝑒))
39		reseq1 5972	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑓 → (𝑣 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐))
40	39	eqeq2d 2774	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → ((𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)))
41	38, 40	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐))))
42	41	cbvralvw 3243	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)))
43	36, 42	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑢 = 𝑒 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐))))
44		fveq1 6880	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑒 → (𝑢‘𝑐) = (𝑒‘𝑐))
45	44	eqeq1d 2765	. . . . . . . . . 10 ⊢ (𝑢 = 𝑒 → ((𝑢‘𝑐) = 𝑎 ↔ (𝑒‘𝑐) = 𝑎))
46	30, 43, 45	3anbi123d 1464	. . . . . . . . 9 ⊢ (𝑢 = 𝑒 → ((𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑎) ↔ (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎)))
47	46	cbvrexvw 3244	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑎) ↔ ∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎))
48	28, 47	bitrdi 290	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (∃𝑢 ∈ 𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥) ↔ ∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎)))
49	48	cbviotavw 6500	. . . . . 6 ⊢ (℩𝑥∃𝑢 ∈ 𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢‘𝑐) = 𝑥)) = (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎))
50	25, 49	eqtrdi 2814	. . . . 5 ⊢ (𝑔 = 𝑐 → (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎)))
51	50	cbvmptv 5215	. . . 4 ⊢ (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = (𝑐 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎)))
52		eleq1w 2846	. . . . . . . . 9 ⊢ (𝑦 = 𝑑 → (𝑦 ∈ dom 𝑢 ↔ 𝑑 ∈ dom 𝑢))
53		suceq 6429	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → suc 𝑦 = suc 𝑑)
54	53	reseq2d 5978	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑑 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑑))
55	53	reseq2d 5978	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑑 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑑))
56	54, 55	eqeq12d 2779	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑑 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)))
57	56	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑦 = 𝑑 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))))
58	57	ralbidv 3188	. . . . . . . . 9 ⊢ (𝑦 = 𝑑 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))))
59	52, 58	anbi12d 643	. . . . . . . 8 ⊢ (𝑦 = 𝑑 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑑 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)))))
60	59	rexbidv 3189	. . . . . . 7 ⊢ (𝑦 = 𝑑 → (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐴 (𝑑 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)))))
61	29	eleq2d 2849	. . . . . . . . 9 ⊢ (𝑢 = 𝑒 → (𝑑 ∈ dom 𝑢 ↔ 𝑑 ∈ dom 𝑒))
62		reseq1 5972	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑒 → (𝑢 ↾ suc 𝑑) = (𝑒 ↾ suc 𝑑))
63	62	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑒 → ((𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑) ↔ (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)))
64	32, 63	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑒 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ (¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))))
65	64	ralbidv 3188	. . . . . . . . . 10 ⊢ (𝑢 = 𝑒 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))))
66		reseq1 5972	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (𝑣 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑))
67	66	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑) ↔ (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))
68	38, 67	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → ((¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑))))
69	68	cbvralvw 3243	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))
70	65, 69	bitrdi 290	. . . . . . . . 9 ⊢ (𝑢 = 𝑒 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑))))
71	61, 70	anbi12d 643	. . . . . . . 8 ⊢ (𝑢 = 𝑒 → ((𝑑 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))) ↔ (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))))
72	71	cbvrexvw 3244	. . . . . . 7 ⊢ (∃𝑢 ∈ 𝐴 (𝑑 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))) ↔ ∃𝑒 ∈ 𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑))))
73	60, 72	bitrdi 290	. . . . . 6 ⊢ (𝑦 = 𝑑 → (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑒 ∈ 𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))))
74	73	cbvabv 2833	. . . . 5 ⊢ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} = {𝑑 ∣ ∃𝑒 ∈ 𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))}
75	74	mpteq1i 5202	. . . 4 ⊢ (𝑐 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎))) = (𝑐 ∈ {𝑑 ∣ ∃𝑒 ∈ 𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎)))
76	51, 75	eqtri 2786	. . 3 ⊢ (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = (𝑐 ∈ {𝑑 ∣ ∃𝑒 ∈ 𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎)))
77	9, 14, 76	ifbieq12i 4515	. 2 ⊢ if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) = if(∃𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏, ((℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2_o⟩}), (𝑐 ∈ {𝑑 ∣ ∃𝑒 ∈ 𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎))))
78	1, 77	eqtri 2786	1 ⊢ 𝑆 = if(∃𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏, ((℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2_o⟩}), (𝑐 ∈ {𝑑 ∣ ∃𝑒 ∈ 𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 {cab 2741 ∀wral 3079 ∃wrex 3089 ∪ cun 3903 ifcif 4487 {csn 4589 ⟨cop 4595 class class class wbr 5109 ↦ cmpt 5192 dom cdm 5661 ↾ cres 5663 suc csuc 6362 ℩cio 6490 ‘cfv 6536 ℩crio 7366 2_oc2o 8443 <s clts 27814

This proof depends on 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-ext 2735

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-sb 2097 df-clab 2742 df-cleq 2755 df-clel 2838 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-mpt 5193 df-xp 5667 df-dm 5671 df-res 5673 df-suc 6366 df-iota 6492 df-fv 6544 df-riota 7367

This theorem is used by: noeta 27916

W3C validator