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Theorem cantnf 8753

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 8737, 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 8742	. 2 ⊢ (𝜑 → 𝑇 Or 𝑆)
6		oecl 7770	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑_𝑜 𝐵) ∈ On)
7	2, 3, 6	syl2anc 565	. . . 4 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ∈ On)
8		eloni 5876	. . . 4 ⊢ ((𝐴 ↑_𝑜 𝐵) ∈ On → Ord (𝐴 ↑_𝑜 𝐵))
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → Ord (𝐴 ↑_𝑜 𝐵))
10		ordwe 5879	. . 3 ⊢ (Ord (𝐴 ↑_𝑜 𝐵) → E We (𝐴 ↑_𝑜 𝐵))
11		weso 5240	. . 3 ⊢ ( E We (𝐴 ↑_𝑜 𝐵) → E Or (𝐴 ↑_𝑜 𝐵))
12		sopo 5187	. . 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 8734	. . 3 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵))
15		frn 6193	. . . . 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 7136	. . . . . . . 8 ⊢ ((𝐴 ↑_𝑜 𝐵) ∈ On → (𝐴 ↑_𝑜 𝐵) ⊆ On)
18	7, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ⊆ On)
19	18	sseld 3749	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ On))
20		eleq1w 2832	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ↔ 𝑦 ∈ (𝐴 ↑_𝑜 𝐵)))
21		eleq1w 2832	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
22	20, 21	imbi12d 333	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → ((𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)) ↔ (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
23	22	imbi2d 329	. . . . . . . 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 3108	. . . . . . . . 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 5882	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Ord (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → 𝑡 ⊆ (𝐴 ↑_𝑜 𝐵))
26	9, 25	sylan 561	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → 𝑡 ⊆ (𝐴 ↑_𝑜 𝐵))
27	26	sselda 3750	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) ∧ 𝑦 ∈ 𝑡) → 𝑦 ∈ (𝐴 ↑_𝑜 𝐵))
28		pm5.5 350	. . . . . . . . . . . . . . . . 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 3133	. . . . . . . . . . . . . . 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 3739	. . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = ∅ → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ ∅ ∈ ran (𝐴 CNF 𝐵)))
34	2	adantr 466	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐴 ∈ On)
35	34	adantr 466	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐴 ∈ On)
36	3	adantr 466	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐵 ∈ On)
37	36	adantr 466	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐵 ∈ On)
38		simplrl 754	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ (𝐴 ↑_𝑜 𝐵))
39		simplrr 755	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ⊆ ran (𝐴 CNF 𝐵))
40	7	adantr 466	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 ↑_𝑜 𝐵) ∈ On)
41		simprl 746	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ (𝐴 ↑_𝑜 𝐵))
42		onelon 5891	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ↑_𝑜 𝐵) ∈ On ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → 𝑡 ∈ On)
43	40, 41, 42	syl2anc 565	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ On)
44		on0eln0 5923	. . . . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → ∅ ∈ 𝑡)
47		eqid 2770	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)} = ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)}
48		eqid 2770	. . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . 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 8752	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
52		fczsupp0 7475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 × {∅}) supp ∅) = ∅
53	52	eqcomi 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ = ((𝐵 × {∅}) supp ∅)
54		oieq2 8573	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ = ((𝐵 × {∅}) supp ∅) → OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅)))
55	53, 54	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅))
56		ne0i 4067	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝐵 → 𝐵 ≠ ∅)
57		ne0i 4067	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → (𝐴 ↑_𝑜 𝐵) ≠ ∅)
58	57	ad2antrl 699	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 ↑_𝑜 𝐵) ≠ ∅)
59		oveq1 6799	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 = ∅ → (𝐴 ↑_𝑜 𝐵) = (∅ ↑_𝑜 𝐵))
60	59	neeq1d 3001	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2965	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((∅ ↑_𝑜 𝐵) = ∅ → 𝐴 ≠ ∅))
63		on0eln0 5923	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
64		oe0m1 7754	. . . . . . . . . . . . . . . . . . . . . . . . . 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 5923	. . . . . . . . . . . . . . . . . . . . . . . . 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 393	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑦 ∈ 𝐵) → ∅ ∈ 𝐴)
72		fconstmpt 5303	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 × {∅}) = (𝑦 ∈ 𝐵 ↦ ∅)
73	71, 72	fmptd 6527	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}):𝐵⟶𝐴)
74		0ex 4921	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∅ ∈ V
75	74	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ∅ ∈ V)
76	3, 75	fczfsuppd 8448	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 × {∅}) finSupp ∅)
77	76	adantr 466	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) finSupp ∅)
78	1, 2, 3	cantnfs 8726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵⟶𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
79	78	adantr 466	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵⟶𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
80	73, 77, 79	mpbir2and 684	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) ∈ 𝑆)
81		eqid 2770	. . . . . . . . . . . . . . . . . . 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 8728	. . . . . . . . . . . . . . . . . 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 5244	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ E We ∅
84		eqid 2770	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ OrdIso( E , ∅) = OrdIso( E , ∅)
85	84	oien 8598	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∅ ∈ V ∧ E We ∅) → dom OrdIso( E , ∅) ≈ ∅)
86	74, 83, 85	mp2an 664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ dom OrdIso( E , ∅) ≈ ∅
87		en0 8171	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (dom OrdIso( E , ∅) ≈ ∅ ↔ dom OrdIso( E , ∅) = ∅)
88	86, 87	mpbi 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom OrdIso( E , ∅) = ∅
89	88	fveq2i 6335	. . . . . . . . . . . . . . . . . . 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 7703	. . . . . . . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
94	14	adantr 466	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵))
95		ffn 6185	. . . . . . . . . . . . . . . . . . 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 6499	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 CNF 𝐵) Fn 𝑆 ∧ (𝐵 × {∅}) ∈ 𝑆) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
98	96, 80, 97	syl2anc 565	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
99	93, 98	eqeltrrd 2850	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ∅ ∈ ran (𝐴 CNF 𝐵))
100	33, 51, 99	pm2.61ne 3027	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
101	100	expr 444	. . . . . . . . . . . . . 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 397	. . . . . . . . . . . 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 7202	. . . . . . 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 3756	. . . 4 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ⊆ ran (𝐴 CNF 𝐵))
112	16, 111	eqssd 3767	. . 3 ⊢ (𝜑 → ran (𝐴 CNF 𝐵) = (𝐴 ↑_𝑜 𝐵))
113		dffo2 6260	. . 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 564	. 2 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆–onto→(𝐴 ↑_𝑜 𝐵))
115	2	adantr 466	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝐴 ∈ On)
116	3	adantr 466	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝐵 ∈ On)
117		fveq2 6332	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → (𝑥‘𝑧) = (𝑥‘𝑡))
118		fveq2 6332	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → (𝑦‘𝑧) = (𝑦‘𝑡))
119	117, 118	eleq12d 2843	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ↔ (𝑥‘𝑡) ∈ (𝑦‘𝑡)))
120		eleq1w 2832	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑡 → (𝑧 ∈ 𝑤 ↔ 𝑡 ∈ 𝑤))
121	120	imbi1d 330	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → ((𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
122	121	ralbidv 3134	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → (∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
123	119, 122	anbi12d 608	. . . . . . . . . 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 3320	. . . . . . . . 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 6331	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑥‘𝑡) = (𝑢‘𝑡))
126		fveq1 6331	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (𝑦‘𝑡) = (𝑣‘𝑡))
127		eleq12 2839	. . . . . . . . . . . 12 ⊢ (((𝑥‘𝑡) = (𝑢‘𝑡) ∧ (𝑦‘𝑡) = (𝑣‘𝑡)) → ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ↔ (𝑢‘𝑡) ∈ (𝑣‘𝑡)))
128	125, 126, 127	syl2an 575	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ↔ (𝑢‘𝑡) ∈ (𝑣‘𝑡)))
129		fveq1 6331	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝑥‘𝑤) = (𝑢‘𝑤))
130		fveq1 6331	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → (𝑦‘𝑤) = (𝑣‘𝑤))
131	129, 130	eqeqan12d 2786	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥‘𝑤) = (𝑦‘𝑤) ↔ (𝑢‘𝑤) = (𝑣‘𝑤)))
132	131	imbi2d 329	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤))))
133	132	ralbidv 3134	. . . . . . . . . . 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 608	. . . . . . . . . 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 3199	. . . . . . . . 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 4854	. . . . . . 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 2792	. . . . . 6 ⊢ 𝑇 = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡 ∈ 𝐵 ((𝑢‘𝑡) ∈ (𝑣‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤)))}
139		simprll 756	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓 ∈ 𝑆)
140		simprlr 757	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑔 ∈ 𝑆)
141		simprr 748	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑇𝑔)
142		eqid 2770	. . . . . 6 ⊢ ∪ {𝑐 ∈ 𝐵 ∣ (𝑓‘𝑐) ∈ (𝑔‘𝑐)} = ∪ {𝑐 ∈ 𝐵 ∣ (𝑓‘𝑐) ∈ (𝑔‘𝑐)}
143		eqid 2770	. . . . . 6 ⊢ OrdIso( E , (𝑔 supp ∅)) = OrdIso( E , (𝑔 supp ∅))
144		eqid 2770	. . . . . 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 8749	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
146		fvex 6342	. . . . . 6 ⊢ ((𝐴 CNF 𝐵)‘𝑔) ∈ V
147	146	epelc 5164	. . . . 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 444	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆)) → (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
150	149	ralrimivva 3119	. 2 ⊢ (𝜑 → ∀𝑓 ∈ 𝑆 ∀𝑔 ∈ 𝑆 (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
151		soisoi 6720	. 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 1476	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 382 = wceq 1630 ∈ wcel 2144 ≠ wne 2942 ∀wral 3060 ∃wrex 3061 {crab 3064 Vcvv 3349 ⊆ wss 3721 ∅c0 4061 {csn 4314 ⟨cop 4320 ∪ cuni 4572 ∩ cint 4609 class class class wbr 4784 {copab 4844 E cep 5161 Po wpo 5168 Or wor 5169 We wwe 5207 × cxp 5247 dom cdm 5249 ran crn 5250 Ord word 5865 Oncon0 5866 ℩cio 5992 Fn wfn 6026 ⟶wf 6027 –onto→wfo 6029 ‘cfv 6031 Isom wiso 6032 (class class class)co 6792 ↦ cmpt2 6794 1^st c1st 7312 2^nd c2nd 7313 supp csupp 7445 seq_𝜔cseqom 7694 +_𝑜 coa 7709 ·_𝑜 comu 7710 ↑_𝑜 coe 7711 ≈ cen 8105 finSupp cfsupp 8430 OrdIsocoi 8569 CNF ccnf 8721

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-supp 7446 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-seqom 7695 df-1o 7712 df-2o 7713 df-oadd 7716 df-omul 7717 df-oexp 7718 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-fsupp 8431 df-oi 8570 df-cnf 8722

This theorem is referenced by: oemapwe 8754 cantnffval2 8755 cantnff1o 8756

W3C validator