cantnf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cantnf		Structured version Visualization version GIF version

Theorem cantnf 8473

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 8457, 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 8462	. 2 ⊢ (𝜑 → 𝑇 Or 𝑆)
6		oecl 7504	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑_𝑜 𝐵) ∈ On)
7	2, 3, 6	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ∈ On)
8		eloni 5650	. . . 4 ⊢ ((𝐴 ↑_𝑜 𝐵) ∈ On → Ord (𝐴 ↑_𝑜 𝐵))
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → Ord (𝐴 ↑_𝑜 𝐵))
10		ordwe 5653	. . 3 ⊢ (Ord (𝐴 ↑_𝑜 𝐵) → E We (𝐴 ↑_𝑜 𝐵))
11		weso 5029	. . 3 ⊢ ( E We (𝐴 ↑_𝑜 𝐵) → E Or (𝐴 ↑_𝑜 𝐵))
12		sopo 4976	. . 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 8454	. . 3 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵))
15		frn 5966	. . . . 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 6882	. . . . . . . 8 ⊢ ((𝐴 ↑_𝑜 𝐵) ∈ On → (𝐴 ↑_𝑜 𝐵) ⊆ On)
18	7, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ⊆ On)
19	18	sseld 3567	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ On))
20		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ↔ 𝑦 ∈ (𝐴 ↑_𝑜 𝐵)))
21		eleq1 2676	. . . . . . . . . 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 2943	. . . . . . . . 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 5656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Ord (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → 𝑡 ⊆ (𝐴 ↑_𝑜 𝐵))
26	9, 25	sylan 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → 𝑡 ⊆ (𝐴 ↑_𝑜 𝐵))
27	26	sselda 3568	. . . . . . . . . . . . . . . . 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 2968	. . . . . . . . . . . . . . 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 3558	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ⊆ ran (𝐴 CNF 𝐵) ↔ ∀𝑦 ∈ 𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵))
32	30, 31	syl6bbr 277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑡 ⊆ ran (𝐴 CNF 𝐵)))
33		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = ∅ → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ ∅ ∈ ran (𝐴 CNF 𝐵)))
34	2	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐴 ∈ On)
35	34	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐴 ∈ On)
36	3	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐵 ∈ On)
37	36	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐵 ∈ On)
38		simplrl 796	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ (𝐴 ↑_𝑜 𝐵))
39		simplrr 797	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ⊆ ran (𝐴 CNF 𝐵))
40	7	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 ↑_𝑜 𝐵) ∈ On)
41		simprl 790	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ (𝐴 ↑_𝑜 𝐵))
42		onelon 5665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ↑_𝑜 𝐵) ∈ On ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → 𝑡 ∈ On)
43	40, 41, 42	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ On)
44		on0eln0 5697	. . . . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → ∅ ∈ 𝑡)
47		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)} = ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_𝑜 𝑐)}
48		eqid 2610	. . . . . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . 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 8472	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
52		fczsupp0 7211	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 × {∅}) supp ∅) = ∅
53	52	eqcomi 2619	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ = ((𝐵 × {∅}) supp ∅)
54		oieq2 8301	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ = ((𝐵 × {∅}) supp ∅) → OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅)))
55	53, 54	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅))
56		ne0i 3880	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝐵 → 𝐵 ≠ ∅)
57		ne0i 3880	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → (𝐴 ↑_𝑜 𝐵) ≠ ∅)
58	57	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 ↑_𝑜 𝐵) ≠ ∅)
59		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 = ∅ → (𝐴 ↑_𝑜 𝐵) = (∅ ↑_𝑜 𝐵))
60	59	neeq1d 2841	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 = ∅ → ((𝐴 ↑_𝑜 𝐵) ≠ ∅ ↔ (∅ ↑_𝑜 𝐵) ≠ ∅))
61	58, 60	syl5ibcom 234	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 = ∅ → (∅ ↑_𝑜 𝐵) ≠ ∅))
62	61	necon2d 2805	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((∅ ↑_𝑜 𝐵) = ∅ → 𝐴 ≠ ∅))
63		on0eln0 5697	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
64		oe0m1 7488	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑_𝑜 𝐵) = ∅))
65	63, 64	bitr3d 269	. . . . . . . . . . . . . . . . . . . . . . . . 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 5697	. . . . . . . . . . . . . . . . . . . . . . . . 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 282	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ → ∅ ∈ 𝐴))
70	56, 69	syl5 33	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝑦 ∈ 𝐵 → ∅ ∈ 𝐴))
71	70	imp 444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑦 ∈ 𝐵) → ∅ ∈ 𝐴)
72		fconstmpt 5085	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 × {∅}) = (𝑦 ∈ 𝐵 ↦ ∅)
73	71, 72	fmptd 6292	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}):𝐵⟶𝐴)
74		0ex 4718	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∅ ∈ V
75	74	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ∅ ∈ V)
76	3, 75	fczfsuppd 8176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 × {∅}) finSupp ∅)
77	76	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) finSupp ∅)
78	1, 2, 3	cantnfs 8446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵⟶𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
79	78	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵⟶𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
80	73, 77, 79	mpbir2and 959	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) ∈ 𝑆)
81		eqid 2610	. . . . . . . . . . . . . . . . . . 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 8448	. . . . . . . . . . . . . . . . . 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 5033	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ E We ∅
84		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ OrdIso( E , ∅) = OrdIso( E , ∅)
85	84	oien 8326	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∅ ∈ V ∧ E We ∅) → dom OrdIso( E , ∅) ≈ ∅)
86	74, 83, 85	mp2an 704	. . . . . . . . . . . . . . . . . . . . 21 ⊢ dom OrdIso( E , ∅) ≈ ∅
87		en0 7905	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (dom OrdIso( E , ∅) ≈ ∅ ↔ dom OrdIso( E , ∅) = ∅)
88	86, 87	mpbi 219	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom OrdIso( E , ∅) = ∅
89	88	fveq2i 6106	. . . . . . . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . . . . . . 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 2632	. . . . . . . . . . . . . . . . . 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 2660	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
94	14	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵))
95		ffn 5958	. . . . . . . . . . . . . . . . . . 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 6264	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 CNF 𝐵) Fn 𝑆 ∧ (𝐵 × {∅}) ∈ 𝑆) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
98	96, 80, 97	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
99	93, 98	eqeltrrd 2689	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ∅ ∈ ran (𝐴 CNF 𝐵))
100	33, 51, 99	pm2.61ne 2867	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
101	100	expr 641	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_𝑜 𝐵)) → (𝑡 ⊆ ran (𝐴 CNF 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
102	32, 101	sylbid 229	. . . . . . . . . . . . 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 449	. . . . . . . . . . . 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 84	. . . . . . . . . . 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 231	. . . . . . . 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 6948	. . . . . . 7 ⊢ (𝑡 ∈ On → (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
109	108	com3l 87	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → (𝑡 ∈ On → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
110	19, 109	mpdd 42	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
111	110	ssrdv 3574	. . . 4 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ⊆ ran (𝐴 CNF 𝐵))
112	16, 111	eqssd 3585	. . 3 ⊢ (𝜑 → ran (𝐴 CNF 𝐵) = (𝐴 ↑_𝑜 𝐵))
113		dffo2 6032	. . 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 695	. 2 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆–onto→(𝐴 ↑_𝑜 𝐵))
115	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝐴 ∈ On)
116	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝐵 ∈ On)
117		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → (𝑥‘𝑧) = (𝑥‘𝑡))
118		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → (𝑦‘𝑧) = (𝑦‘𝑡))
119	117, 118	eleq12d 2682	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ↔ (𝑥‘𝑡) ∈ (𝑦‘𝑡)))
120		eleq1 2676	. . . . . . . . . . . . 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 2969	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → (∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
123	119, 122	anbi12d 743	. . . . . . . . . 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 3148	. . . . . . . . 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 6102	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑥‘𝑡) = (𝑢‘𝑡))
126		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (𝑦‘𝑡) = (𝑣‘𝑡))
127		eleq12 2678	. . . . . . . . . . . 12 ⊢ (((𝑥‘𝑡) = (𝑢‘𝑡) ∧ (𝑦‘𝑡) = (𝑣‘𝑡)) → ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ↔ (𝑢‘𝑡) ∈ (𝑣‘𝑡)))
128	125, 126, 127	syl2an 493	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ↔ (𝑢‘𝑡) ∈ (𝑣‘𝑡)))
129		fveq1 6102	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝑥‘𝑤) = (𝑢‘𝑤))
130		fveq1 6102	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → (𝑦‘𝑤) = (𝑣‘𝑤))
131	129, 130	eqeqan12d 2626	. . . . . . . . . . . . 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 2969	. . . . . . . . . . 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 743	. . . . . . . . . 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 3034	. . . . . . . . 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 271	. . . . . . . 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 4654	. . . . . . 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 2632	. . . . . 6 ⊢ 𝑇 = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡 ∈ 𝐵 ((𝑢‘𝑡) ∈ (𝑣‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤)))}
139		simprll 798	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓 ∈ 𝑆)
140		simprlr 799	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑔 ∈ 𝑆)
141		simprr 792	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑇𝑔)
142		eqid 2610	. . . . . 6 ⊢ ∪ {𝑐 ∈ 𝐵 ∣ (𝑓‘𝑐) ∈ (𝑔‘𝑐)} = ∪ {𝑐 ∈ 𝐵 ∣ (𝑓‘𝑐) ∈ (𝑔‘𝑐)}
143		eqid 2610	. . . . . 6 ⊢ OrdIso( E , (𝑔 supp ∅)) = OrdIso( E , (𝑔 supp ∅))
144		eqid 2610	. . . . . 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 8469	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
146		fvex 6113	. . . . . 6 ⊢ ((𝐴 CNF 𝐵)‘𝑔) ∈ V
147	146	epelc 4951	. . . . 5 ⊢ (((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔) ↔ ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
148	145, 147	sylibr 223	. . . 4 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔))
149	148	expr 641	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆)) → (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
150	149	ralrimivva 2954	. 2 ⊢ (𝜑 → ∀𝑓 ∈ 𝑆 ∀𝑔 ∈ 𝑆 (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
151		soisoi 6478	. 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 1319	1 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑_𝑜 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 {csn 4125 ⟨cop 4131 ∪ cuni 4372 ∩ cint 4410 class class class wbr 4583 {copab 4642 E cep 4947 Po wpo 4957 Or wor 4958 We wwe 4996 × cxp 5036 dom cdm 5038 ran crn 5039 Ord word 5639 Oncon0 5640 ℩cio 5766 Fn wfn 5799 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058 supp csupp 7182 seq_𝜔cseqom 7429 +_𝑜 coa 7444 ·_𝑜 comu 7445 ↑_𝑜 coe 7446 ≈ cen 7838 finSupp cfsupp 8158 OrdIsocoi 8297 CNF ccnf 8441

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-oexp 7453 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-cnf 8442

This theorem is referenced by: oemapwe 8474 cantnffval2 8475 cantnff1o 8476

W3C validator