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Theorem cantnf 8763
Description: The Cantor Normal Form theorem. The function (𝐴 CNF 𝐵), which maps a finitely supported function from 𝐵 to 𝐴 to the sum ((𝐴𝑜 𝑓(𝑎1)) ∘ 𝑎1) +𝑜 ((𝐴𝑜 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 (𝐴𝑜 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 8747, 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 (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
Assertion
Ref Expression
cantnf (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴𝑜 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.
StepHypRef Expression
1 cantnfs.s . . 3 𝑆 = dom (𝐴 CNF 𝐵)
2 cantnfs.a . . 3 (𝜑𝐴 ∈ On)
3 cantnfs.b . . 3 (𝜑𝐵 ∈ On)
4 oemapval.t . . 3 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
51, 2, 3, 4oemapso 8752 . 2 (𝜑𝑇 Or 𝑆)
6 oecl 7786 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
72, 3, 6syl2anc 696 . . . 4 (𝜑 → (𝐴𝑜 𝐵) ∈ On)
8 eloni 5894 . . . 4 ((𝐴𝑜 𝐵) ∈ On → Ord (𝐴𝑜 𝐵))
97, 8syl 17 . . 3 (𝜑 → Ord (𝐴𝑜 𝐵))
10 ordwe 5897 . . 3 (Ord (𝐴𝑜 𝐵) → E We (𝐴𝑜 𝐵))
11 weso 5257 . . 3 ( E We (𝐴𝑜 𝐵) → E Or (𝐴𝑜 𝐵))
12 sopo 5204 . . 3 ( E Or (𝐴𝑜 𝐵) → E Po (𝐴𝑜 𝐵))
139, 10, 11, 124syl 19 . 2 (𝜑 → E Po (𝐴𝑜 𝐵))
141, 2, 3cantnff 8744 . . 3 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵))
15 frn 6214 . . . . 5 ((𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵) → ran (𝐴 CNF 𝐵) ⊆ (𝐴𝑜 𝐵))
1614, 15syl 17 . . . 4 (𝜑 → ran (𝐴 CNF 𝐵) ⊆ (𝐴𝑜 𝐵))
17 onss 7155 . . . . . . . 8 ((𝐴𝑜 𝐵) ∈ On → (𝐴𝑜 𝐵) ⊆ On)
187, 17syl 17 . . . . . . 7 (𝜑 → (𝐴𝑜 𝐵) ⊆ On)
1918sseld 3743 . . . . . 6 (𝜑 → (𝑡 ∈ (𝐴𝑜 𝐵) → 𝑡 ∈ On))
20 eleq1w 2822 . . . . . . . . . 10 (𝑡 = 𝑦 → (𝑡 ∈ (𝐴𝑜 𝐵) ↔ 𝑦 ∈ (𝐴𝑜 𝐵)))
21 eleq1w 2822 . . . . . . . . . 10 (𝑡 = 𝑦 → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
2220, 21imbi12d 333 . . . . . . . . 9 (𝑡 = 𝑦 → ((𝑡 ∈ (𝐴𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)) ↔ (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
2322imbi2d 329 . . . . . . . 8 (𝑡 = 𝑦 → ((𝜑 → (𝑡 ∈ (𝐴𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)))))
24 r19.21v 3098 . . . . . . . . 9 (∀𝑦𝑡 (𝜑 → (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → ∀𝑦𝑡 (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
25 ordelss 5900 . . . . . . . . . . . . . . . . . . 19 ((Ord (𝐴𝑜 𝐵) ∧ 𝑡 ∈ (𝐴𝑜 𝐵)) → 𝑡 ⊆ (𝐴𝑜 𝐵))
269, 25sylan 489 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 ∈ (𝐴𝑜 𝐵)) → 𝑡 ⊆ (𝐴𝑜 𝐵))
2726sselda 3744 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡 ∈ (𝐴𝑜 𝐵)) ∧ 𝑦𝑡) → 𝑦 ∈ (𝐴𝑜 𝐵))
28 pm5.5 350 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (𝐴𝑜 𝐵) → ((𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
2927, 28syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑡 ∈ (𝐴𝑜 𝐵)) ∧ 𝑦𝑡) → ((𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
3029ralbidva 3123 . . . . . . . . . . . . . . 15 ((𝜑𝑡 ∈ (𝐴𝑜 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ ∀𝑦𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵)))
31 dfss3 3733 . . . . . . . . . . . . . . 15 (𝑡 ⊆ ran (𝐴 CNF 𝐵) ↔ ∀𝑦𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵))
3230, 31syl6bbr 278 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴𝑜 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑡 ⊆ ran (𝐴 CNF 𝐵)))
33 eleq1 2827 . . . . . . . . . . . . . . . 16 (𝑡 = ∅ → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ ∅ ∈ ran (𝐴 CNF 𝐵)))
342adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐴 ∈ On)
3534adantr 472 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐴 ∈ On)
363adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐵 ∈ On)
3736adantr 472 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐵 ∈ On)
38 simplrl 819 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ (𝐴𝑜 𝐵))
39 simplrr 820 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ⊆ ran (𝐴 CNF 𝐵))
407adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴𝑜 𝐵) ∈ On)
41 simprl 811 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ (𝐴𝑜 𝐵))
42 onelon 5909 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑜 𝐵) ∈ On ∧ 𝑡 ∈ (𝐴𝑜 𝐵)) → 𝑡 ∈ On)
4340, 41, 42syl2anc 696 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ On)
44 on0eln0 5941 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ On → (∅ ∈ 𝑡𝑡 ≠ ∅))
4543, 44syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝑡𝑡 ≠ ∅))
4645biimpar 503 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → ∅ ∈ 𝑡)
47 eqid 2760 . . . . . . . . . . . . . . . . 17 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)} = {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)}
48 eqid 2760 . . . . . . . . . . . . . . . . 17 (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)}) ·𝑜 𝑎) +𝑜 𝑏) = 𝑡)) = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)}) ·𝑜 𝑎) +𝑜 𝑏) = 𝑡))
49 eqid 2760 . . . . . . . . . . . . . . . . 17 (1st ‘(℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)}) ·𝑜 𝑎) +𝑜 𝑏) = 𝑡))) = (1st ‘(℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)}) ·𝑜 𝑎) +𝑜 𝑏) = 𝑡)))
50 eqid 2760 . . . . . . . . . . . . . . . . 17 (2nd ‘(℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)}) ·𝑜 𝑎) +𝑜 𝑏) = 𝑡))) = (2nd ‘(℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)})(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴𝑜 𝑐)}) ·𝑜 𝑎) +𝑜 𝑏) = 𝑡)))
511, 35, 37, 4, 38, 39, 46, 47, 48, 49, 50cantnflem4 8762 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
52 fczsupp0 7493 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 × {∅}) supp ∅) = ∅
5352eqcomi 2769 . . . . . . . . . . . . . . . . . . . 20 ∅ = ((𝐵 × {∅}) supp ∅)
54 oieq2 8583 . . . . . . . . . . . . . . . . . . . 20 (∅ = ((𝐵 × {∅}) supp ∅) → OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅)))
5553, 54ax-mp 5 . . . . . . . . . . . . . . . . . . 19 OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅))
56 ne0i 4064 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝐵𝐵 ≠ ∅)
57 ne0i 4064 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 ∈ (𝐴𝑜 𝐵) → (𝐴𝑜 𝐵) ≠ ∅)
5857ad2antrl 766 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴𝑜 𝐵) ≠ ∅)
59 oveq1 6820 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
6059neeq1d 2991 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ≠ ∅ ↔ (∅ ↑𝑜 𝐵) ≠ ∅))
6158, 60syl5ibcom 235 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 = ∅ → (∅ ↑𝑜 𝐵) ≠ ∅))
6261necon2d 2955 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((∅ ↑𝑜 𝐵) = ∅ → 𝐴 ≠ ∅))
63 on0eln0 5941 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
64 oe0m1 7770 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
6563, 64bitr3d 270 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 ∈ On → (𝐵 ≠ ∅ ↔ (∅ ↑𝑜 𝐵) = ∅))
6636, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ ↔ (∅ ↑𝑜 𝐵) = ∅))
67 on0eln0 5941 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
6834, 67syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝐴𝐴 ≠ ∅))
6962, 66, 683imtr4d 283 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ → ∅ ∈ 𝐴))
7056, 69syl5 34 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝑦𝐵 → ∅ ∈ 𝐴))
7170imp 444 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑦𝐵) → ∅ ∈ 𝐴)
72 fconstmpt 5320 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 × {∅}) = (𝑦𝐵 ↦ ∅)
7371, 72fmptd 6548 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}):𝐵𝐴)
74 0ex 4942 . . . . . . . . . . . . . . . . . . . . . . 23 ∅ ∈ V
7574a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ∅ ∈ V)
763, 75fczfsuppd 8458 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐵 × {∅}) finSupp ∅)
7776adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) finSupp ∅)
781, 2, 3cantnfs 8736 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
7978adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
8073, 77, 79mpbir2and 995 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) ∈ 𝑆)
81 eqid 2760 . . . . . . . . . . . . . . . . . . 19 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ∅)‘𝑘)) ·𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ∅)‘𝑘)) ·𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +𝑜 𝑧)), ∅)
821, 34, 36, 55, 80, 81cantnfval 8738 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ∅)‘𝑘)) ·𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +𝑜 𝑧)), ∅)‘dom OrdIso( E , ∅)))
83 we0 5261 . . . . . . . . . . . . . . . . . . . . . 22 E We ∅
84 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . 23 OrdIso( E , ∅) = OrdIso( E , ∅)
8584oien 8608 . . . . . . . . . . . . . . . . . . . . . 22 ((∅ ∈ V ∧ E We ∅) → dom OrdIso( E , ∅) ≈ ∅)
8674, 83, 85mp2an 710 . . . . . . . . . . . . . . . . . . . . 21 dom OrdIso( E , ∅) ≈ ∅
87 en0 8184 . . . . . . . . . . . . . . . . . . . . 21 (dom OrdIso( E , ∅) ≈ ∅ ↔ dom OrdIso( E , ∅) = ∅)
8886, 87mpbi 220 . . . . . . . . . . . . . . . . . . . 20 dom OrdIso( E , ∅) = ∅
8988fveq2i 6355 . . . . . . . . . . . . . . . . . . 19 (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ∅)‘𝑘)) ·𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +𝑜 𝑧)), ∅)‘dom OrdIso( E , ∅)) = (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ∅)‘𝑘)) ·𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +𝑜 𝑧)), ∅)‘∅)
9081seqom0g 7720 . . . . . . . . . . . . . . . . . . . 20 (∅ ∈ V → (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ∅)‘𝑘)) ·𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +𝑜 𝑧)), ∅)‘∅) = ∅)
9174, 90ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ∅)‘𝑘)) ·𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +𝑜 𝑧)), ∅)‘∅) = ∅
9289, 91eqtri 2782 . . . . . . . . . . . . . . . . . 18 (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ∅)‘𝑘)) ·𝑜 ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +𝑜 𝑧)), ∅)‘dom OrdIso( E , ∅)) = ∅
9382, 92syl6eq 2810 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
9414adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵))
95 ffn 6206 . . . . . . . . . . . . . . . . . . 19 ((𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
9694, 95syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵) Fn 𝑆)
97 fnfvelrn 6519 . . . . . . . . . . . . . . . . . 18 (((𝐴 CNF 𝐵) Fn 𝑆 ∧ (𝐵 × {∅}) ∈ 𝑆) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
9896, 80, 97syl2anc 696 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
9993, 98eqeltrrd 2840 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ∅ ∈ ran (𝐴 CNF 𝐵))
10033, 51, 99pm2.61ne 3017 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡 ∈ (𝐴𝑜 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
101100expr 644 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴𝑜 𝐵)) → (𝑡 ⊆ ran (𝐴 CNF 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
10232, 101sylbid 230 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ (𝐴𝑜 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
103102ex 449 . . . . . . . . . . . 12 (𝜑 → (𝑡 ∈ (𝐴𝑜 𝐵) → (∀𝑦𝑡 (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
104103com23 86 . . . . . . . . . . 11 (𝜑 → (∀𝑦𝑡 (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → (𝑡 ∈ (𝐴𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
105104a2i 14 . . . . . . . . . 10 ((𝜑 → ∀𝑦𝑡 (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
106105a1i 11 . . . . . . . . 9 (𝑡 ∈ On → ((𝜑 → ∀𝑦𝑡 (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))))
10724, 106syl5bi 232 . . . . . . . 8 (𝑡 ∈ On → (∀𝑦𝑡 (𝜑 → (𝑦 ∈ (𝐴𝑜 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))))
10823, 107tfis2 7221 . . . . . . 7 (𝑡 ∈ On → (𝜑 → (𝑡 ∈ (𝐴𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
109108com3l 89 . . . . . 6 (𝜑 → (𝑡 ∈ (𝐴𝑜 𝐵) → (𝑡 ∈ On → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
11019, 109mpdd 43 . . . . 5 (𝜑 → (𝑡 ∈ (𝐴𝑜 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
111110ssrdv 3750 . . . 4 (𝜑 → (𝐴𝑜 𝐵) ⊆ ran (𝐴 CNF 𝐵))
11216, 111eqssd 3761 . . 3 (𝜑 → ran (𝐴 CNF 𝐵) = (𝐴𝑜 𝐵))
113 dffo2 6280 . . 3 ((𝐴 CNF 𝐵):𝑆onto→(𝐴𝑜 𝐵) ↔ ((𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵) ∧ ran (𝐴 CNF 𝐵) = (𝐴𝑜 𝐵)))
11414, 112, 113sylanbrc 701 . 2 (𝜑 → (𝐴 CNF 𝐵):𝑆onto→(𝐴𝑜 𝐵))
1152adantr 472 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝐴 ∈ On)
1163adantr 472 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝐵 ∈ On)
117 fveq2 6352 . . . . . . . . . . . 12 (𝑧 = 𝑡 → (𝑥𝑧) = (𝑥𝑡))
118 fveq2 6352 . . . . . . . . . . . 12 (𝑧 = 𝑡 → (𝑦𝑧) = (𝑦𝑡))
119117, 118eleq12d 2833 . . . . . . . . . . 11 (𝑧 = 𝑡 → ((𝑥𝑧) ∈ (𝑦𝑧) ↔ (𝑥𝑡) ∈ (𝑦𝑡)))
120 eleq1w 2822 . . . . . . . . . . . . 13 (𝑧 = 𝑡 → (𝑧𝑤𝑡𝑤))
121120imbi1d 330 . . . . . . . . . . . 12 (𝑧 = 𝑡 → ((𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
122121ralbidv 3124 . . . . . . . . . . 11 (𝑧 = 𝑡 → (∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
123119, 122anbi12d 749 . . . . . . . . . 10 (𝑧 = 𝑡 → (((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
124123cbvrexv 3311 . . . . . . . . 9 (∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
125 fveq1 6351 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (𝑥𝑡) = (𝑢𝑡))
126 fveq1 6351 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (𝑦𝑡) = (𝑣𝑡))
127 eleq12 2829 . . . . . . . . . . . 12 (((𝑥𝑡) = (𝑢𝑡) ∧ (𝑦𝑡) = (𝑣𝑡)) → ((𝑥𝑡) ∈ (𝑦𝑡) ↔ (𝑢𝑡) ∈ (𝑣𝑡)))
128125, 126, 127syl2an 495 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝑡) ∈ (𝑦𝑡) ↔ (𝑢𝑡) ∈ (𝑣𝑡)))
129 fveq1 6351 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (𝑥𝑤) = (𝑢𝑤))
130 fveq1 6351 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 → (𝑦𝑤) = (𝑣𝑤))
131129, 130eqeqan12d 2776 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝑢𝑤) = (𝑣𝑤)))
132131imbi2d 329 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤))))
133132ralbidv 3124 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → (∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤))))
134128, 133anbi12d 749 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑣) → (((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
135134rexbidv 3190 . . . . . . . . 9 ((𝑥 = 𝑢𝑦 = 𝑣) → (∃𝑡𝐵 ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
136124, 135syl5bb 272 . . . . . . . 8 ((𝑥 = 𝑢𝑦 = 𝑣) → (∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
137136cbvopabv 4874 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))}
1384, 137eqtri 2782 . . . . . 6 𝑇 = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))}
139 simprll 821 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑆)
140 simprlr 822 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑔𝑆)
141 simprr 813 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑇𝑔)
142 eqid 2760 . . . . . 6 {𝑐𝐵 ∣ (𝑓𝑐) ∈ (𝑔𝑐)} = {𝑐𝐵 ∣ (𝑓𝑐) ∈ (𝑔𝑐)}
143 eqid 2760 . . . . . 6 OrdIso( E , (𝑔 supp ∅)) = OrdIso( E , (𝑔 supp ∅))
144 eqid 2760 . . . . . 6 seq𝜔((𝑘 ∈ V, 𝑡 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , (𝑔 supp ∅))‘𝑘)) ·𝑜 (𝑔‘(OrdIso( E , (𝑔 supp ∅))‘𝑘))) +𝑜 𝑡)), ∅) = seq𝜔((𝑘 ∈ V, 𝑡 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , (𝑔 supp ∅))‘𝑘)) ·𝑜 (𝑔‘(OrdIso( E , (𝑔 supp ∅))‘𝑘))) +𝑜 𝑡)), ∅)
1451, 115, 116, 138, 139, 140, 141, 142, 143, 144cantnflem1 8759 . . . . 5 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
146 fvex 6362 . . . . . 6 ((𝐴 CNF 𝐵)‘𝑔) ∈ V
147146epelc 5181 . . . . 5 (((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔) ↔ ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
148145, 147sylibr 224 . . . 4 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔))
149148expr 644 . . 3 ((𝜑 ∧ (𝑓𝑆𝑔𝑆)) → (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
150149ralrimivva 3109 . 2 (𝜑 → ∀𝑓𝑆𝑔𝑆 (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
151 soisoi 6741 . 2 (((𝑇 Or 𝑆 ∧ E Po (𝐴𝑜 𝐵)) ∧ ((𝐴 CNF 𝐵):𝑆onto→(𝐴𝑜 𝐵) ∧ ∀𝑓𝑆𝑔𝑆 (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))) → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴𝑜 𝐵)))
1525, 13, 114, 150, 151syl22anc 1478 1 (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  wss 3715  c0 4058  {csn 4321  cop 4327   cuni 4588   cint 4627   class class class wbr 4804  {copab 4864   E cep 5178   Po wpo 5185   Or wor 5186   We wwe 5224   × cxp 5264  dom cdm 5266  ran crn 5267  Ord word 5883  Oncon0 5884  cio 6010   Fn wfn 6044  wf 6045  ontowfo 6047  cfv 6049   Isom wiso 6050  (class class class)co 6813  cmpt2 6815  1st c1st 7331  2nd c2nd 7332   supp csupp 7463  seq𝜔cseqom 7711   +𝑜 coa 7726   ·𝑜 comu 7727  𝑜 coe 7728  cen 8118   finSupp cfsupp 8440  OrdIsocoi 8579   CNF ccnf 8731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-seqom 7712  df-1o 7729  df-2o 7730  df-oadd 7733  df-omul 7734  df-oexp 7735  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-oi 8580  df-cnf 8732
This theorem is referenced by:  oemapwe  8764  cantnffval2  8765  cantnff1o  8766
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