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Theorem cantnf 8587

Description: The Cantor Normal Form theorem. The function (𝐴 CNF 𝐵), which maps a finitely supported function from 𝐵 to 𝐴 to the sum ((𝐴 ↑_𝑜 𝑓(𝑎1)) ∘ 𝑎1) +_𝑜 ((𝐴 ↑_𝑜 𝑓(𝑎2)) ∘ 𝑎2) +_𝑜 ... over all indexes 𝑎 < 𝐵 such that 𝑓(𝑎) is nonzero, is an order isomorphism from the ordering 𝑇 of finitely supported functions to the set (𝐴 ↑_𝑜 𝐵) under the natural order. Setting 𝐴 = ω and letting 𝐵 be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 8571, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.)

**Hypotheses**
Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
oemapval.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}

**Assertion**
Ref	Expression
cantnf	⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑_𝑜 𝐵)))

Distinct variable groups: 𝑥,𝑤,𝑦,𝑧,𝐵 𝑤,𝐴,𝑥,𝑦,𝑧 𝑥,𝑆,𝑦,𝑧 𝜑,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑤) 𝑆(𝑤) 𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cantnf Dummy variables 𝑓 𝑐 𝑔 𝑘 𝑡 𝑢 𝑣 𝑎 𝑏 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfs.s	. . 3 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
2		cantnfs.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
3		cantnfs.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
4		oemapval.t	. . 3 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
5	1, 2, 3, 4	oemapso 8576	. 2 ⊢ (𝜑 → 𝑇 Or 𝑆)
6		oecl 7614	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑_𝑜 𝐵) ∈ On)
7	2, 3, 6	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ∈ On)
8		eloni 5731	. . . 4 ⊢ ((𝐴 ↑_𝑜 𝐵) ∈ On → Ord (𝐴 ↑_𝑜 𝐵))
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → Ord (𝐴 ↑_𝑜 𝐵))
10		ordwe 5734	. . 3 ⊢ (Ord (𝐴 ↑_𝑜 𝐵) → E We (𝐴 ↑_𝑜 𝐵))
11		weso 5103	. . 3 ⊢ ( E We (𝐴 ↑_𝑜 𝐵) → E Or (𝐴 ↑_𝑜 𝐵))
12		sopo 5050	. . 3 ⊢ ( E Or (𝐴 ↑_𝑜 𝐵) → E Po (𝐴 ↑_𝑜 𝐵))
13	9, 10, 11, 12	4syl 19	. 2 ⊢ (𝜑 → E Po (𝐴 ↑_𝑜 𝐵))
14	1, 2, 3	cantnff 8568	. . 3 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵))
15		frn 6051	. . . . 5 ⊢ ((𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵) → ran (𝐴 CNF 𝐵) ⊆ (𝐴 ↑_𝑜 𝐵))
16	14, 15	syl 17	. . . 4 ⊢ (𝜑 → ran (𝐴 CNF 𝐵) ⊆ (𝐴 ↑_𝑜 𝐵))
17		onss 6987	. . . . . . . 8 ⊢ ((𝐴 ↑_𝑜 𝐵) ∈ On → (𝐴 ↑_𝑜 𝐵) ⊆ On)
18	7, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ⊆ On)
19	18	sseld 3600	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ On))
20		eleq1 2688	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ↔ 𝑦 ∈ (𝐴 ↑_𝑜 𝐵)))
21		eleq1 2688	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
22	20, 21	imbi12d 334	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → ((𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)) ↔ (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
23	22	imbi2d 330	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ((𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)))))
24		r19.21v 2959	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑡 (𝜑 → (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → ∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
25		ordelss 5737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Ord (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → 𝑡 ⊆ (𝐴 ↑_𝑜 𝐵))
26	9, 25	sylan 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → 𝑡 ⊆ (𝐴 ↑_𝑜 𝐵))
27	26	sselda 3601	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) ∧ 𝑦 ∈ 𝑡) → 𝑦 ∈ (𝐴 ↑_𝑜 𝐵))
28		pm5.5 351	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → ((𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
29	27, 28	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) ∧ 𝑦 ∈ 𝑡) → ((𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
30	29	ralbidva 2984	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ ∀𝑦 ∈ 𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵)))
31		dfss3 3590	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ⊆ ran (𝐴 CNF 𝐵) ↔ ∀𝑦 ∈ 𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵))
32	30, 31	syl6bbr 278	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑡 ⊆ ran (𝐴 CNF 𝐵)))
33		eleq1 2688	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = ∅ → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ ∅ ∈ ran (𝐴 CNF 𝐵)))
34	2	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐴 ∈ On)
35	34	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐴 ∈ On)
36	3	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐵 ∈ On)
37	36	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐵 ∈ On)
38		simplrl 800	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ (𝐴 ↑_𝑜 𝐵))
39		simplrr 801	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ⊆ ran (𝐴 CNF 𝐵))
40	7	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 ↑_𝑜 𝐵) ∈ On)
41		simprl 794	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ (𝐴 ↑_𝑜 𝐵))
42		onelon 5746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ↑_𝑜 𝐵) ∈ On ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → 𝑡 ∈ On)
43	40, 41, 42	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ On)
44		on0eln0 5778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ On → (∅ ∈ 𝑡 ↔ 𝑡 ≠ ∅))
45	43, 44	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝑡 ↔ 𝑡 ≠ ∅))
46	45	biimpar 502	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → ∅ ∈ 𝑡)
47		eqid 2621	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)} = ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)}
48		eqid 2621	. . . . . . . . . . . . . . . . 17 ⊢ (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)}) ·_𝑜 𝑎) +_𝑜 𝑏) = 𝑡)) = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)}) ·_𝑜 𝑎) +_𝑜 𝑏) = 𝑡))
49		eqid 2621	. . . . . . . . . . . . . . . . 17 ⊢ (1^st ‘(℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)}) ·_𝑜 𝑎) +_𝑜 𝑏) = 𝑡))) = (1^st ‘(℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)}) ·_𝑜 𝑎) +_𝑜 𝑏) = 𝑡)))
50		eqid 2621	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘(℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)}) ·_𝑜 𝑎) +_𝑜 𝑏) = 𝑡))) = (2^nd ‘(℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑_𝑜 ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)}) ·_𝑜 𝑎) +_𝑜 𝑏) = 𝑡)))
51	1, 35, 37, 4, 38, 39, 46, 47, 48, 49, 50	cantnflem4 8586	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
52		fczsupp0 7321	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 × {∅}) supp ∅) = ∅
53	52	eqcomi 2630	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ = ((𝐵 × {∅}) supp ∅)
54		oieq2 8415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ = ((𝐵 × {∅}) supp ∅) → OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅)))
55	53, 54	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅))
56		ne0i 3919	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝐵 → 𝐵 ≠ ∅)
57		ne0i 3919	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → (𝐴 ↑_𝑜 𝐵) ≠ ∅)
58	57	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 ↑_𝑜 𝐵) ≠ ∅)
59		oveq1 6654	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 = ∅ → (𝐴 ↑_𝑜 𝐵) = (∅ ↑_𝑜 𝐵))
60	59	neeq1d 2852	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 = ∅ → ((𝐴 ↑_𝑜 𝐵) ≠ ∅ ↔ (∅ ↑_𝑜 𝐵) ≠ ∅))
61	58, 60	syl5ibcom 235	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 = ∅ → (∅ ↑_𝑜 𝐵) ≠ ∅))
62	61	necon2d 2816	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((∅ ↑_𝑜 𝐵) = ∅ → 𝐴 ≠ ∅))
63		on0eln0 5778	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
64		oe0m1 7598	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑_𝑜 𝐵) = ∅))
65	63, 64	bitr3d 270	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐵 ∈ On → (𝐵 ≠ ∅ ↔ (∅ ↑_𝑜 𝐵) = ∅))
66	36, 65	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ ↔ (∅ ↑_𝑜 𝐵) = ∅))
67		on0eln0 5778	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
68	34, 67	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
69	62, 66, 68	3imtr4d 283	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ → ∅ ∈ 𝐴))
70	56, 69	syl5 34	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝑦 ∈ 𝐵 → ∅ ∈ 𝐴))
71	70	imp 445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑦 ∈ 𝐵) → ∅ ∈ 𝐴)
72		fconstmpt 5161	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 × {∅}) = (𝑦 ∈ 𝐵 ↦ ∅)
73	71, 72	fmptd 6383	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}):𝐵⟶𝐴)
74		0ex 4788	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∅ ∈ V
75	74	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ∅ ∈ V)
76	3, 75	fczfsuppd 8290	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 × {∅}) finSupp ∅)
77	76	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) finSupp ∅)
78	1, 2, 3	cantnfs 8560	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵⟶𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
79	78	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵⟶𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
80	73, 77, 79	mpbir2and 957	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) ∈ 𝑆)
81		eqid 2621	. . . . . . . . . . . . . . . . . . 19 ⊢ seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , ∅)‘𝑘)) ·_𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_𝑜 𝑧)), ∅) = seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , ∅)‘𝑘)) ·_𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_𝑜 𝑧)), ∅)
82	1, 34, 36, 55, 80, 81	cantnfval 8562	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = (seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , ∅)‘𝑘)) ·_𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_𝑜 𝑧)), ∅)‘dom OrdIso( E , ∅)))
83		we0 5107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ E We ∅
84		eqid 2621	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ OrdIso( E , ∅) = OrdIso( E , ∅)
85	84	oien 8440	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∅ ∈ V ∧ E We ∅) → dom OrdIso( E , ∅) ≈ ∅)
86	74, 83, 85	mp2an 708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ dom OrdIso( E , ∅) ≈ ∅
87		en0 8016	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (dom OrdIso( E , ∅) ≈ ∅ ↔ dom OrdIso( E , ∅) = ∅)
88	86, 87	mpbi 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom OrdIso( E , ∅) = ∅
89	88	fveq2i 6192	. . . . . . . . . . . . . . . . . . 19 ⊢ (seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , ∅)‘𝑘)) ·_𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_𝑜 𝑧)), ∅)‘dom OrdIso( E , ∅)) = (seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , ∅)‘𝑘)) ·_𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_𝑜 𝑧)), ∅)‘∅)
90	81	seqom0g 7548	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ ∈ V → (seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , ∅)‘𝑘)) ·_𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_𝑜 𝑧)), ∅)‘∅) = ∅)
91	74, 90	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , ∅)‘𝑘)) ·_𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_𝑜 𝑧)), ∅)‘∅) = ∅
92	89, 91	eqtri 2643	. . . . . . . . . . . . . . . . . 18 ⊢ (seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , ∅)‘𝑘)) ·_𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_𝑜 𝑧)), ∅)‘dom OrdIso( E , ∅)) = ∅
93	82, 92	syl6eq 2671	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
94	14	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵))
95		ffn 6043	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
96	94, 95	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵) Fn 𝑆)
97		fnfvelrn 6354	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 CNF 𝐵) Fn 𝑆 ∧ (𝐵 × {∅}) ∈ 𝑆) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
98	96, 80, 97	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
99	93, 98	eqeltrrd 2701	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ∅ ∈ ran (𝐴 CNF 𝐵))
100	33, 51, 99	pm2.61ne 2878	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
101	100	expr 643	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → (𝑡 ⊆ ran (𝐴 CNF 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
102	32, 101	sylbid 230	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
103	102	ex 450	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
104	103	com23 86	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
105	104	a2i 14	. . . . . . . . . 10 ⊢ ((𝜑 → ∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
106	105	a1i 11	. . . . . . . . 9 ⊢ (𝑡 ∈ On → ((𝜑 → ∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))))
107	24, 106	syl5bi 232	. . . . . . . 8 ⊢ (𝑡 ∈ On → (∀𝑦 ∈ 𝑡 (𝜑 → (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))))
108	23, 107	tfis2 7053	. . . . . . 7 ⊢ (𝑡 ∈ On → (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
109	108	com3l 89	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → (𝑡 ∈ On → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
110	19, 109	mpdd 43	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
111	110	ssrdv 3607	. . . 4 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ⊆ ran (𝐴 CNF 𝐵))
112	16, 111	eqssd 3618	. . 3 ⊢ (𝜑 → ran (𝐴 CNF 𝐵) = (𝐴 ↑_𝑜 𝐵))
113		dffo2 6117	. . 3 ⊢ ((𝐴 CNF 𝐵):𝑆–onto→(𝐴 ↑_𝑜 𝐵) ↔ ((𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵) ∧ ran (𝐴 CNF 𝐵) = (𝐴 ↑_𝑜 𝐵)))
114	14, 112, 113	sylanbrc 698	. 2 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆–onto→(𝐴 ↑_𝑜 𝐵))
115	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝐴 ∈ On)
116	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝐵 ∈ On)
117		fveq2 6189	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → (𝑥‘𝑧) = (𝑥‘𝑡))
118		fveq2 6189	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → (𝑦‘𝑧) = (𝑦‘𝑡))
119	117, 118	eleq12d 2694	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ↔ (𝑥‘𝑡) ∈ (𝑦‘𝑡)))
120		eleq1 2688	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑡 → (𝑧 ∈ 𝑤 ↔ 𝑡 ∈ 𝑤))
121	120	imbi1d 331	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → ((𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
122	121	ralbidv 2985	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → (∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
123	119, 122	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑧 = 𝑡 → (((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))))
124	123	cbvrexv 3170	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑡 ∈ 𝐵 ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
125		fveq1 6188	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑥‘𝑡) = (𝑢‘𝑡))
126		fveq1 6188	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (𝑦‘𝑡) = (𝑣‘𝑡))
127		eleq12 2690	. . . . . . . . . . . 12 ⊢ (((𝑥‘𝑡) = (𝑢‘𝑡) ∧ (𝑦‘𝑡) = (𝑣‘𝑡)) → ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ↔ (𝑢‘𝑡) ∈ (𝑣‘𝑡)))
128	125, 126, 127	syl2an 494	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ↔ (𝑢‘𝑡) ∈ (𝑣‘𝑡)))
129		fveq1 6188	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝑥‘𝑤) = (𝑢‘𝑤))
130		fveq1 6188	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → (𝑦‘𝑤) = (𝑣‘𝑤))
131	129, 130	eqeqan12d 2637	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥‘𝑤) = (𝑦‘𝑤) ↔ (𝑢‘𝑤) = (𝑣‘𝑤)))
132	131	imbi2d 330	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤))))
133	132	ralbidv 2985	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤))))
134	128, 133	anbi12d 747	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (((𝑥‘𝑡) ∈ (𝑦‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ((𝑢‘𝑡) ∈ (𝑣‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤)))))
135	134	rexbidv 3050	. . . . . . . . 9 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (∃𝑡 ∈ 𝐵 ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑡 ∈ 𝐵 ((𝑢‘𝑡) ∈ (𝑣‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤)))))
136	124, 135	syl5bb 272	. . . . . . . 8 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑡 ∈ 𝐵 ((𝑢‘𝑡) ∈ (𝑣‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤)))))
137	136	cbvopabv 4720	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡 ∈ 𝐵 ((𝑢‘𝑡) ∈ (𝑣‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤)))}
138	4, 137	eqtri 2643	. . . . . 6 ⊢ 𝑇 = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡 ∈ 𝐵 ((𝑢‘𝑡) ∈ (𝑣‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤)))}
139		simprll 802	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓 ∈ 𝑆)
140		simprlr 803	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑔 ∈ 𝑆)
141		simprr 796	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑇𝑔)
142		eqid 2621	. . . . . 6 ⊢ ∪ {𝑐 ∈ 𝐵 ∣ (𝑓‘𝑐) ∈ (𝑔‘𝑐)} = ∪ {𝑐 ∈ 𝐵 ∣ (𝑓‘𝑐) ∈ (𝑔‘𝑐)}
143		eqid 2621	. . . . . 6 ⊢ OrdIso( E , (𝑔 supp ∅)) = OrdIso( E , (𝑔 supp ∅))
144		eqid 2621	. . . . . 6 ⊢ seq_𝜔((𝑘 ∈ V, 𝑡 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , (𝑔 supp ∅))‘𝑘)) ·_𝑜 (𝑔‘(OrdIso( E , (𝑔 supp ∅))‘𝑘))) +_𝑜 𝑡)), ∅) = seq_𝜔((𝑘 ∈ V, 𝑡 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , (𝑔 supp ∅))‘𝑘)) ·_𝑜 (𝑔‘(OrdIso( E , (𝑔 supp ∅))‘𝑘))) +_𝑜 𝑡)), ∅)
145	1, 115, 116, 138, 139, 140, 141, 142, 143, 144	cantnflem1 8583	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
146		fvex 6199	. . . . . 6 ⊢ ((𝐴 CNF 𝐵)‘𝑔) ∈ V
147	146	epelc 5029	. . . . 5 ⊢ (((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔) ↔ ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
148	145, 147	sylibr 224	. . . 4 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔))
149	148	expr 643	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆)) → (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
150	149	ralrimivva 2970	. 2 ⊢ (𝜑 → ∀𝑓 ∈ 𝑆 ∀𝑔 ∈ 𝑆 (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
151		soisoi 6575	. 2 ⊢ (((𝑇 Or 𝑆 ∧ E Po (𝐴 ↑_𝑜 𝐵)) ∧ ((𝐴 CNF 𝐵):𝑆–onto→(𝐴 ↑_𝑜 𝐵) ∧ ∀𝑓 ∈ 𝑆 ∀𝑔 ∈ 𝑆 (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))) → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑_𝑜 𝐵)))
152	5, 13, 114, 150, 151	syl22anc 1326	1 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑_𝑜 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1482 ∈ wcel 1989 ≠ wne 2793 ∀wral 2911 ∃wrex 2912 {crab 2915 Vcvv 3198 ⊆ wss 3572 ∅c0 3913 {csn 4175 ⟨cop 4181 ∪ cuni 4434 ∩ cint 4473 class class class wbr 4651 {copab 4710 E cep 5026 Po wpo 5031 Or wor 5032 We wwe 5070 × cxp 5110 dom cdm 5112 ran crn 5113 Ord word 5720 Oncon0 5721 ℩cio 5847 Fn wfn 5881 ⟶wf 5882 –onto→wfo 5884 ‘cfv 5886 Isom wiso 5887 (class class class)co 6647 ↦ cmpt2 6649 1^st c1st 7163 2^nd c2nd 7164 supp csupp 7292 seq_𝜔cseqom 7539 +_𝑜 coa 7554 ·_𝑜 comu 7555 ↑_𝑜 coe 7556 ≈ cen 7949 finSupp cfsupp 8272 OrdIsocoi 8411 CNF ccnf 8555

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1721 ax-4 1736 ax-5 1838 ax-6 1887 ax-7 1934 ax-8 1991 ax-9 1998 ax-10 2018 ax-11 2033 ax-12 2046 ax-13 2245 ax-ext 2601 ax-rep 4769 ax-sep 4779 ax-nul 4787 ax-pow 4841 ax-pr 4904 ax-un 6946

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1485 df-fal 1488 df-ex 1704 df-nf 1709 df-sb 1880 df-eu 2473 df-mo 2474 df-clab 2608 df-cleq 2614 df-clel 2617 df-nfc 2752 df-ne 2794 df-ral 2916 df-rex 2917 df-reu 2918 df-rmo 2919 df-rab 2920 df-v 3200 df-sbc 3434 df-csb 3532 df-dif 3575 df-un 3577 df-in 3579 df-ss 3586 df-pss 3588 df-nul 3914 df-if 4085 df-pw 4158 df-sn 4176 df-pr 4178 df-tp 4180 df-op 4182 df-uni 4435 df-int 4474 df-iun 4520 df-br 4652 df-opab 4711 df-mpt 4728 df-tr 4751 df-id 5022 df-eprel 5027 df-po 5033 df-so 5034 df-fr 5071 df-se 5072 df-we 5073 df-xp 5118 df-rel 5119 df-cnv 5120 df-co 5121 df-dm 5122 df-rn 5123 df-res 5124 df-ima 5125 df-pred 5678 df-ord 5724 df-on 5725 df-lim 5726 df-suc 5727 df-iota 5849 df-fun 5888 df-fn 5889 df-f 5890 df-f1 5891 df-fo 5892 df-f1o 5893 df-fv 5894 df-isom 5895 df-riota 6608 df-ov 6650 df-oprab 6651 df-mpt2 6652 df-om 7063 df-1st 7165 df-2nd 7166 df-supp 7293 df-wrecs 7404 df-recs 7465 df-rdg 7503 df-seqom 7540 df-1o 7557 df-2o 7558 df-oadd 7561 df-omul 7562 df-oexp 7563 df-er 7739 df-map 7856 df-en 7953 df-dom 7954 df-sdom 7955 df-fin 7956 df-fsupp 8273 df-oi 8412 df-cnf 8556

This theorem is referenced by: oemapwe 8588 cantnffval2 8589 cantnff1o 8590

W3C validator