Theorem nosupcbv 33950

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), 2o⟩}), (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), 2o⟩}), (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,𝑏   𝑎,𝑐,𝐴   𝐴,𝑑   𝑒,𝑎,𝐴   𝐴,𝑓   𝑔,𝑎,𝐴,𝑢,𝑣   𝑥,𝑎,𝐴,𝑦,𝑏   𝑐,𝑑,𝑒,𝑓   𝑔,𝑐,𝑢,𝑣   𝑥,𝑐,𝑦   𝑒,𝑑,𝑓,𝑢,𝑣,𝑦   𝑒,𝑔,𝑢,𝑣   𝑥,𝑒,𝑦   𝑓,𝑔,𝑢,𝑣   𝑥,𝑓,𝑦   𝑢,𝑔,𝑣   𝑥,𝑔,𝑦   𝑣,𝑢   𝑥,𝑢,𝑦   𝑦,𝑣   𝑥,𝑦

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), 2o⟩}), (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 5084	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 <s 𝑦 ↔ 𝑎 <s 𝑦))
3	2	notbid 318	. . . . . 6 ⊢ (𝑥 = 𝑎 → (¬ 𝑥 <s 𝑦 ↔ ¬ 𝑎 <s 𝑦))
4	3	ralbidv 3171	. . . . 5 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑎 <s 𝑦))
5		breq2 5085	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑎 <s 𝑦 ↔ 𝑎 <s 𝑏))
6	5	notbid 318	. . . . . 6 ⊢ (𝑦 = 𝑏 → (¬ 𝑎 <s 𝑦 ↔ ¬ 𝑎 <s 𝑏))
7	6	cbvralvw 3222	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑎 <s 𝑦 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏)
8	4, 7	bitrdi 287	. . . 4 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏))
9	8	cbvrexvw 3223	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ ∃𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏)
10	8	cbvriotavw 7274	. . . 4 ⊢ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏)
11	10	dmeqi 5826	. . . . . 6 ⊢ dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏)
12	11	opeq1i 4812	. . . . 5 ⊢ ⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩ = ⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2o⟩
13	12	sneqi 4576	. . . 4 ⊢ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩} = {⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2o⟩}
14	10, 13	uneq12i 4101	. . 3 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) = ((℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2o⟩})
15		eleq1w 2819	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → (𝑔 ∈ dom 𝑢 ↔ 𝑐 ∈ dom 𝑢))
16		suceq 6346	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑐 → suc 𝑔 = suc 𝑐)
17	16	reseq2d 5903	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑐 → (𝑢 ↾ suc 𝑔) = (𝑢 ↾ suc 𝑐))
18	16	reseq2d 5903	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑐 → (𝑣 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑐))
19	17, 18	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑐 → ((𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔) ↔ (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
20	19	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑔 = 𝑐 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
21	20	ralbidv 3171	. . . . . . . . 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 6813	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑐) = 𝑥))
23	15, 21, 22	3anbi123d 1436	. . . . . . . 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 3172	. . . . . . 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 6442	. . . . . 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 2748	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((𝑢‘𝑐) = 𝑥 ↔ (𝑢‘𝑐) = 𝑎))
27	26	3anbi3d 1442	. . . . . . . . 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 3172	. . . . . . . 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 5825	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑒 → dom 𝑢 = dom 𝑒)
30	29	eleq2d 2822	. . . . . . . . . 10 ⊢ (𝑢 = 𝑒 → (𝑐 ∈ dom 𝑢 ↔ 𝑐 ∈ dom 𝑒))
31		breq2 5085	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑒 → (𝑣 <s 𝑢 ↔ 𝑣 <s 𝑒))
32	31	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑒 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑒))
33		reseq1 5897	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑒 → (𝑢 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))
34	33	eqeq1d 2738	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑒 → ((𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
35	32, 34	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑒 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
36	35	ralbidv 3171	. . . . . . . . . . 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 5084	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑓 → (𝑣 <s 𝑒 ↔ 𝑓 <s 𝑒))
38	37	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (¬ 𝑣 <s 𝑒 ↔ ¬ 𝑓 <s 𝑒))
39		reseq1 5897	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑓 → (𝑣 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐))
40	39	eqeq2d 2747	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → ((𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)))
41	38, 40	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐))))
42	41	cbvralvw 3222	. . . . . . . . . . 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 287	. . . . . . . . . 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 6803	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑒 → (𝑢‘𝑐) = (𝑒‘𝑐))
45	44	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑢 = 𝑒 → ((𝑢‘𝑐) = 𝑎 ↔ (𝑒‘𝑐) = 𝑎))
46	30, 43, 45	3anbi123d 1436	. . . . . . . . 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 3223	. . . . . . . 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 287	. . . . . . 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 6418	. . . . . 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 2792	. . . . 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 5194	. . . 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 2819	. . . . . . . . 9 ⊢ (𝑦 = 𝑑 → (𝑦 ∈ dom 𝑢 ↔ 𝑑 ∈ dom 𝑢))
53		suceq 6346	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → suc 𝑦 = suc 𝑑)
54	53	reseq2d 5903	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑑 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑑))
55	53	reseq2d 5903	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑑 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑑))
56	54, 55	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑑 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)))
57	56	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑦 = 𝑑 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))))
58	57	ralbidv 3171	. . . . . . . . 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 632	. . . . . . . 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 3172	. . . . . . 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 2822	. . . . . . . . 9 ⊢ (𝑢 = 𝑒 → (𝑑 ∈ dom 𝑢 ↔ 𝑑 ∈ dom 𝑒))
62		reseq1 5897	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑒 → (𝑢 ↾ suc 𝑑) = (𝑒 ↾ suc 𝑑))
63	62	eqeq1d 2738	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑒 → ((𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑) ↔ (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)))
64	32, 63	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑒 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ (¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))))
65	64	ralbidv 3171	. . . . . . . . . 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 5897	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (𝑣 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑))
67	66	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑) ↔ (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))
68	38, 67	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → ((¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑))))
69	68	cbvralvw 3222	. . . . . . . . . 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 287	. . . . . . . . 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 632	. . . . . . . 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 3223	. . . . . . 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 287	. . . . . 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 2809	. . . . 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 5177	. . . 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 2764	. . 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 4492	. 2 ⊢ if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (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), 2o⟩}), (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 2764	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), 2o⟩}), (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))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  {cab 2713  ∀wral 3062  ∃wrex 3071   ∪ cun 3890  ifcif 4465  {csn 4565  ⟨cop 4571   class class class wbr 5081   ↦ cmpt 5164  dom cdm 5600   ↾ cres 5602  suc csuc 6283  ℩cio 6408  ‘cfv 6458  ℩crio 7263  2_oc2o 8322   <s cslt 33889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-xp 5606  df-dm 5610  df-res 5612  df-suc 6287  df-iota 6410  df-fv 6466  df-riota 7264

This theorem is referenced by:  noeta  33991