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Theorem cantnf 9672

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 indices 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 9656, 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 9661	. 2 ⊢ (𝜑 → 𝑇 Or 𝑆)
6		oecl 8531	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑_o 𝐵) ∈ On)
7	2, 3, 6	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐴 ↑_o 𝐵) ∈ On)
8		eloni 6377	. . . 4 ⊢ ((𝐴 ↑_o 𝐵) ∈ On → Ord (𝐴 ↑_o 𝐵))
9	7, 8	syl 18	. . 3 ⊢ (𝜑 → Ord (𝐴 ↑_o 𝐵))
10		ordwe 6380	. . 3 ⊢ (Ord (𝐴 ↑_o 𝐵) → E We (𝐴 ↑_o 𝐵))
11		weso 5657	. . 3 ⊢ ( E We (𝐴 ↑_o 𝐵) → E Or (𝐴 ↑_o 𝐵))
12		sopo 5593	. . 3 ⊢ ( E Or (𝐴 ↑_o 𝐵) → E Po (𝐴 ↑_o 𝐵))
13	9, 10, 11, 12	4syl 20	. 2 ⊢ (𝜑 → E Po (𝐴 ↑_o 𝐵))
14	1, 2, 3	cantnff 9653	. . 3 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_o 𝐵))
15	14	frnd 6721	. . . 4 ⊢ (𝜑 → ran (𝐴 CNF 𝐵) ⊆ (𝐴 ↑_o 𝐵))
16		onss 7793	. . . . . . . 8 ⊢ ((𝐴 ↑_o 𝐵) ∈ On → (𝐴 ↑_o 𝐵) ⊆ On)
17	7, 16	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑_o 𝐵) ⊆ On)
18	17	sseld 3939	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_o 𝐵) → 𝑡 ∈ On))
19		eleq1w 2849	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ (𝐴 ↑_o 𝐵) ↔ 𝑦 ∈ (𝐴 ↑_o 𝐵)))
20		eleq1w 2849	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
21	19, 20	imbi12d 347	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → ((𝑡 ∈ (𝐴 ↑_o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)) ↔ (𝑦 ∈ (𝐴 ↑_o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
22	21	imbi2d 343	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ((𝜑 → (𝑡 ∈ (𝐴 ↑_o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → (𝑦 ∈ (𝐴 ↑_o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)))))
23		r19.21v 3193	. . . . . . . . 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))))
24		ordelss 6383	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Ord (𝐴 ↑_o 𝐵) ∧ 𝑡 ∈ (𝐴 ↑_o 𝐵)) → 𝑡 ⊆ (𝐴 ↑_o 𝐵))
25	9, 24	sylan 592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_o 𝐵)) → 𝑡 ⊆ (𝐴 ↑_o 𝐵))
26	25	sselda 3940	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_o 𝐵)) ∧ 𝑦 ∈ 𝑡) → 𝑦 ∈ (𝐴 ↑_o 𝐵))
27		pm5.5 364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴 ↑_o 𝐵) → ((𝑦 ∈ (𝐴 ↑_o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
28	26, 27	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_o 𝐵)) ∧ 𝑦 ∈ 𝑡) → ((𝑦 ∈ (𝐴 ↑_o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
29	28	ralbidva 3189	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_o 𝐵)) → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ ∀𝑦 ∈ 𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵)))
30		dfss3 3929	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ⊆ ran (𝐴 CNF 𝐵) ↔ ∀𝑦 ∈ 𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵))
31	29, 30	bitr4di 292	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_o 𝐵)) → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑡 ⊆ ran (𝐴 CNF 𝐵)))
32		eleq1 2854	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = ∅ → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ ∅ ∈ ran (𝐴 CNF 𝐵)))
33	2	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐴 ∈ On)
34	33	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐴 ∈ On)
35	3	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐵 ∈ On)
36	35	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐵 ∈ On)
37		simplrl 789	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ (𝐴 ↑_o 𝐵))
38		simplrr 790	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ⊆ ran (𝐴 CNF 𝐵))
39	7	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 ↑_o 𝐵) ∈ On)
40		simprl 783	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ (𝐴 ↑_o 𝐵))
41		onelon 6392	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ↑_o 𝐵) ∈ On ∧ 𝑡 ∈ (𝐴 ↑_o 𝐵)) → 𝑡 ∈ On)
42	39, 40, 41	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ On)
43		on0eln0 6425	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ On → (∅ ∈ 𝑡 ↔ 𝑡 ≠ ∅))
44	42, 43	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝑡 ↔ 𝑡 ≠ ∅))
45	44	biimpar 483	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → ∅ ∈ 𝑡)
46		eqid 2766	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_o 𝑐)} = ∪ ∩ {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴 ↑_o 𝑐)}
47		eqid 2766	. . . . . . . . . . . . . . . . 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))
48		eqid 2766	. . . . . . . . . . . . . . . . 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)))
49		eqid 2766	. . . . . . . . . . . . . . . . 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)))
50	1, 34, 36, 4, 37, 38, 45, 46, 47, 48, 49	cantnflem4 9671	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
51		fczsupp0 8198	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 × {∅}) supp ∅) = ∅
52	51	eqcomi 2775	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ = ((𝐵 × {∅}) supp ∅)
53		oieq2 9485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ = ((𝐵 × {∅}) supp ∅) → OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅)))
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅))
55		ne0i 4297	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ (𝐴 ↑_o 𝐵) → (𝐴 ↑_o 𝐵) ≠ ∅)
56	55	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 ↑_o 𝐵) ≠ ∅)
57		oveq1 7430	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 = ∅ → (𝐴 ↑_o 𝐵) = (∅ ↑_o 𝐵))
58	57	neeq1d 3020	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 = ∅ → ((𝐴 ↑_o 𝐵) ≠ ∅ ↔ (∅ ↑_o 𝐵) ≠ ∅))
59	56, 58	syl5ibcom 248	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 = ∅ → (∅ ↑_o 𝐵) ≠ ∅))
60	59	necon2d 2984	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((∅ ↑_o 𝐵) = ∅ → 𝐴 ≠ ∅))
61		on0eln0 6425	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
62		oe0m1 8515	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑_o 𝐵) = ∅))
63	61, 62	bitr3d 284	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐵 ∈ On → (𝐵 ≠ ∅ ↔ (∅ ↑_o 𝐵) = ∅))
64	35, 63	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ ↔ (∅ ↑_o 𝐵) = ∅))
65		on0eln0 6425	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
66	33, 65	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
67	60, 64, 66	3imtr4d 297	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ → ∅ ∈ 𝐴))
68		ne0i 4297	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝐵 → 𝐵 ≠ ∅)
69	67, 68	impel 515	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑦 ∈ 𝐵) → ∅ ∈ 𝐴)
70		fconstmpt 5728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 × {∅}) = (𝑦 ∈ 𝐵 ↦ ∅)
71	69, 70	fmptd 7116	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}):𝐵⟶𝐴)
72		0ex 5275	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∅ ∈ V
73	72	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ∅ ∈ V)
74	3, 73	fczfsuppd 9356	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 × {∅}) finSupp ∅)
75	74	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) finSupp ∅)
76	1, 2, 3	cantnfs 9645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵⟶𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
77	76	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵⟶𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
78	71, 75, 77	mpbir2and 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) ∈ 𝑆)
79		eqid 2766	. . . . . . . . . . . . . . . . . . 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)), ∅)
80	1, 33, 35, 54, 78, 79	cantnfval 9647	. . . . . . . . . . . . . . . . . 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 , ∅)))
81		we0 5661	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ E We ∅
82		eqid 2766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ OrdIso( E , ∅) = OrdIso( E , ∅)
83	82	oien 9510	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∅ ∈ V ∧ E We ∅) → dom OrdIso( E , ∅) ≈ ∅)
84	72, 81, 83	mp2an 705	. . . . . . . . . . . . . . . . . . . . 21 ⊢ dom OrdIso( E , ∅) ≈ ∅
85		en0 9024	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (dom OrdIso( E , ∅) ≈ ∅ ↔ dom OrdIso( E , ∅) = ∅)
86	84, 85	mpbi 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom OrdIso( E , ∅) = ∅
87	86	fveq2i 6891	. . . . . . . . . . . . . . . . . . 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)), ∅)‘∅)
88	79	seqom0g 8452	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ ∈ V → (seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_o (OrdIso( E , ∅)‘𝑘)) ·_o ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_o 𝑧)), ∅)‘∅) = ∅)
89	72, 88	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_o (OrdIso( E , ∅)‘𝑘)) ·_o ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_o 𝑧)), ∅)‘∅) = ∅
90	87, 89	eqtri 2789	. . . . . . . . . . . . . . . . . 18 ⊢ (seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_o (OrdIso( E , ∅)‘𝑘)) ·_o ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +_o 𝑧)), ∅)‘dom OrdIso( E , ∅)) = ∅
91	80, 90	eqtrdi 2817	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
92	14	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_o 𝐵))
93	92	ffnd 6713	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵) Fn 𝑆)
94		fnfvelrn 7082	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 CNF 𝐵) Fn 𝑆 ∧ (𝐵 × {∅}) ∈ 𝑆) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
95	93, 78, 94	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
96	91, 95	eqeltrrd 2867	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ∅ ∈ ran (𝐴 CNF 𝐵))
97	32, 50, 96	pm2.61ne 3046	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝐴 ↑_o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
98	97	expr 462	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_o 𝐵)) → (𝑡 ⊆ ran (𝐴 CNF 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
99	31, 98	sylbid 243	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴 ↑_o 𝐵)) → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
100	99	ex 418	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_o 𝐵) → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
101	100	com23 87	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → (𝑡 ∈ (𝐴 ↑_o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
102	101	a2i 15	. . . . . . . . . 10 ⊢ ((𝜑 → ∀𝑦 ∈ 𝑡 (𝑦 ∈ (𝐴 ↑_o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴 ↑_o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
103	102	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)))))
104	23, 103	biimtrid 245	. . . . . . . 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)))))
105	22, 104	tfis2 7862	. . . . . . 7 ⊢ (𝑡 ∈ On → (𝜑 → (𝑡 ∈ (𝐴 ↑_o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
106	105	com3l 90	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_o 𝐵) → (𝑡 ∈ On → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
107	18, 106	mpdd 44	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝐴 ↑_o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
108	107	ssrdv 3946	. . . 4 ⊢ (𝜑 → (𝐴 ↑_o 𝐵) ⊆ ran (𝐴 CNF 𝐵))
109	15, 108	eqssd 3957	. . 3 ⊢ (𝜑 → ran (𝐴 CNF 𝐵) = (𝐴 ↑_o 𝐵))
110		dffo2 6803	. . 3 ⊢ ((𝐴 CNF 𝐵):𝑆–onto→(𝐴 ↑_o 𝐵) ↔ ((𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_o 𝐵) ∧ ran (𝐴 CNF 𝐵) = (𝐴 ↑_o 𝐵)))
111	14, 109, 110	sylanbrc 595	. 2 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆–onto→(𝐴 ↑_o 𝐵))
112	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝐴 ∈ On)
113	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝐵 ∈ On)
114		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → (𝑥‘𝑧) = (𝑥‘𝑡))
115		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → (𝑦‘𝑧) = (𝑦‘𝑡))
116	114, 115	eleq12d 2860	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ↔ (𝑥‘𝑡) ∈ (𝑦‘𝑡)))
117		eleq1w 2849	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑡 → (𝑧 ∈ 𝑤 ↔ 𝑡 ∈ 𝑤))
118	117	imbi1d 344	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → ((𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
119	118	ralbidv 3191	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → (∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))
120	116, 119	anbi12d 644	. . . . . . . . . 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)))))
121	120	cbvrexvw 3247	. . . . . . . . 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))))
122		fveq1 6887	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑥‘𝑡) = (𝑢‘𝑡))
123		fveq1 6887	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (𝑦‘𝑡) = (𝑣‘𝑡))
124		eleq12 2856	. . . . . . . . . . . 12 ⊢ (((𝑥‘𝑡) = (𝑢‘𝑡) ∧ (𝑦‘𝑡) = (𝑣‘𝑡)) → ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ↔ (𝑢‘𝑡) ∈ (𝑣‘𝑡)))
125	122, 123, 124	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥‘𝑡) ∈ (𝑦‘𝑡) ↔ (𝑢‘𝑡) ∈ (𝑣‘𝑡)))
126		fveq1 6887	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝑥‘𝑤) = (𝑢‘𝑤))
127		fveq1 6887	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → (𝑦‘𝑤) = (𝑣‘𝑤))
128	126, 127	eqeqan12d 2780	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥‘𝑤) = (𝑦‘𝑤) ↔ (𝑢‘𝑤) = (𝑣‘𝑤)))
129	128	imbi2d 343	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤))))
130	129	ralbidv 3191	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤))))
131	125, 130	anbi12d 644	. . . . . . . . . 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)))))
132	131	rexbidv 3192	. . . . . . . . 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)))))
133	121, 132	bitrid 286	. . . . . . . 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)))))
134	133	cbvopabv 5189	. . . . . . 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)))}
135	4, 134	eqtri 2789	. . . . . 6 ⊢ 𝑇 = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡 ∈ 𝐵 ((𝑢‘𝑡) ∈ (𝑣‘𝑡) ∧ ∀𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 → (𝑢‘𝑤) = (𝑣‘𝑤)))}
136		simprll 791	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓 ∈ 𝑆)
137		simprlr 792	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑔 ∈ 𝑆)
138		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑇𝑔)
139		eqid 2766	. . . . . 6 ⊢ ∪ {𝑐 ∈ 𝐵 ∣ (𝑓‘𝑐) ∈ (𝑔‘𝑐)} = ∪ {𝑐 ∈ 𝐵 ∣ (𝑓‘𝑐) ∈ (𝑔‘𝑐)}
140		eqid 2766	. . . . . 6 ⊢ OrdIso( E , (𝑔 supp ∅)) = OrdIso( E , (𝑔 supp ∅))
141		eqid 2766	. . . . . 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)), ∅)
142	1, 112, 113, 135, 136, 137, 138, 139, 140, 141	cantnflem1 9668	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
143		fvex 6901	. . . . . 6 ⊢ ((𝐴 CNF 𝐵)‘𝑔) ∈ V
144	143	epeli 5568	. . . . 5 ⊢ (((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔) ↔ ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
145	142, 144	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔))
146	145	expr 462	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆)) → (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
147	146	ralrimivva 3211	. 2 ⊢ (𝜑 → ∀𝑓 ∈ 𝑆 ∀𝑔 ∈ 𝑆 (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
148		soisoi 7337	. 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)))
149	5, 13, 111, 147, 148	syl22anc 852	1 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑_o 𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 ∃wrex 3092 {crab 3419 Vcvv 3458 ⊆ wss 3908 ∅c0 4289 {csn 4594 ⟨cop 4600 ∪ cuni 4877 ∩ cint 4917 class class class wbr 5114 {copab 5178 E cep 5565 Po wpo 5572 Or wor 5573 We wwe 5618 × cxp 5664 dom cdm 5666 ran crn 5667 Ord word 6366 Oncon0 6367 ℩cio 6497 Fn wfn 6538 ⟶wf 6539 –onto→wfo 6541 ‘cfv 6543 Isom wiso 6544 (class class class)co 7423 ∈ cmpo 7425 1^st c1st 7993 2^nd c2nd 7994 supp csupp 8165 seq_ωcseqom 8443 +_o coa 8459 ·_o comu 8460 ↑_o coe 8461 ≈ cen 8949 finSupp cfsupp 9331 OrdIsocoi 9481 CNF ccnf 9640

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-seqom 8444 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-oexp 8468 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-oi 9482 df-cnf 9641

This theorem is used by: oemapwe 9673 cantnffval2 9674 cantnff1o 9675 cantnfresb 44091

W3C validator