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Theorem cantnf 8948
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 indices 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 8932, 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 8937 . 2 (𝜑𝑇 Or 𝑆)
6 oecl 7962 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
72, 3, 6syl2anc 576 . . . 4 (𝜑 → (𝐴o 𝐵) ∈ On)
8 eloni 6036 . . . 4 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
97, 8syl 17 . . 3 (𝜑 → Ord (𝐴o 𝐵))
10 ordwe 6039 . . 3 (Ord (𝐴o 𝐵) → E We (𝐴o 𝐵))
11 weso 5394 . . 3 ( E We (𝐴o 𝐵) → E Or (𝐴o 𝐵))
12 sopo 5340 . . 3 ( E Or (𝐴o 𝐵) → E Po (𝐴o 𝐵))
139, 10, 11, 124syl 19 . 2 (𝜑 → E Po (𝐴o 𝐵))
141, 2, 3cantnff 8929 . . 3 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
1514frnd 6348 . . . 4 (𝜑 → ran (𝐴 CNF 𝐵) ⊆ (𝐴o 𝐵))
16 onss 7319 . . . . . . . 8 ((𝐴o 𝐵) ∈ On → (𝐴o 𝐵) ⊆ On)
177, 16syl 17 . . . . . . 7 (𝜑 → (𝐴o 𝐵) ⊆ On)
1817sseld 3850 . . . . . 6 (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ On))
19 eleq1w 2841 . . . . . . . . . 10 (𝑡 = 𝑦 → (𝑡 ∈ (𝐴o 𝐵) ↔ 𝑦 ∈ (𝐴o 𝐵)))
20 eleq1w 2841 . . . . . . . . . 10 (𝑡 = 𝑦 → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
2119, 20imbi12d 337 . . . . . . . . 9 (𝑡 = 𝑦 → ((𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)) ↔ (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
2221imbi2d 333 . . . . . . . 8 (𝑡 = 𝑦 → ((𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)))))
23 r19.21v 3118 . . . . . . . . 9 (∀𝑦𝑡 (𝜑 → (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → ∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
24 ordelss 6042 . . . . . . . . . . . . . . . . . . 19 ((Ord (𝐴o 𝐵) ∧ 𝑡 ∈ (𝐴o 𝐵)) → 𝑡 ⊆ (𝐴o 𝐵))
259, 24sylan 572 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → 𝑡 ⊆ (𝐴o 𝐵))
2625sselda 3851 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡 ∈ (𝐴o 𝐵)) ∧ 𝑦𝑡) → 𝑦 ∈ (𝐴o 𝐵))
27 pm5.5 354 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (𝐴o 𝐵) → ((𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
2826, 27syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑡 ∈ (𝐴o 𝐵)) ∧ 𝑦𝑡) → ((𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
2928ralbidva 3139 . . . . . . . . . . . . . . 15 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ ∀𝑦𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵)))
30 dfss3 3840 . . . . . . . . . . . . . . 15 (𝑡 ⊆ ran (𝐴 CNF 𝐵) ↔ ∀𝑦𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵))
3129, 30syl6bbr 281 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑡 ⊆ ran (𝐴 CNF 𝐵)))
32 eleq1 2846 . . . . . . . . . . . . . . . 16 (𝑡 = ∅ → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ ∅ ∈ ran (𝐴 CNF 𝐵)))
332adantr 473 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐴 ∈ On)
3433adantr 473 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐴 ∈ On)
353adantr 473 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐵 ∈ On)
3635adantr 473 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐵 ∈ On)
37 simplrl 765 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ (𝐴o 𝐵))
38 simplrr 766 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ⊆ ran (𝐴 CNF 𝐵))
397adantr 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴o 𝐵) ∈ On)
40 simprl 759 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ (𝐴o 𝐵))
41 onelon 6051 . . . . . . . . . . . . . . . . . . . 20 (((𝐴o 𝐵) ∈ On ∧ 𝑡 ∈ (𝐴o 𝐵)) → 𝑡 ∈ On)
4239, 40, 41syl2anc 576 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ On)
43 on0eln0 6081 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ On → (∅ ∈ 𝑡𝑡 ≠ ∅))
4442, 43syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝑡𝑡 ≠ ∅))
4544biimpar 470 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → ∅ ∈ 𝑡)
46 eqid 2771 . . . . . . . . . . . . . . . . 17 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴o 𝑐)} = {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴o 𝑐)}
47 eqid 2771 . . . . . . . . . . . . . . . . 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))
48 eqid 2771 . . . . . . . . . . . . . . . . 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)))
49 eqid 2771 . . . . . . . . . . . . . . . . 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)))
501, 34, 36, 4, 37, 38, 45, 46, 47, 48, 49cantnflem4 8947 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
51 fczsupp0 7660 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 × {∅}) supp ∅) = ∅
5251eqcomi 2780 . . . . . . . . . . . . . . . . . . . 20 ∅ = ((𝐵 × {∅}) supp ∅)
53 oieq2 8770 . . . . . . . . . . . . . . . . . . . 20 (∅ = ((𝐵 × {∅}) supp ∅) → OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅)))
5452, 53ax-mp 5 . . . . . . . . . . . . . . . . . . 19 OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅))
55 ne0i 4180 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 ∈ (𝐴o 𝐵) → (𝐴o 𝐵) ≠ ∅)
5655ad2antrl 716 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴o 𝐵) ≠ ∅)
57 oveq1 6981 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
5857neeq1d 3019 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = ∅ → ((𝐴o 𝐵) ≠ ∅ ↔ (∅ ↑o 𝐵) ≠ ∅))
5956, 58syl5ibcom 237 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 = ∅ → (∅ ↑o 𝐵) ≠ ∅))
6059necon2d 2983 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((∅ ↑o 𝐵) = ∅ → 𝐴 ≠ ∅))
61 on0eln0 6081 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
62 oe0m1 7946 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
6361, 62bitr3d 273 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 ∈ On → (𝐵 ≠ ∅ ↔ (∅ ↑o 𝐵) = ∅))
6435, 63syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ ↔ (∅ ↑o 𝐵) = ∅))
65 on0eln0 6081 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
6633, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝐴𝐴 ≠ ∅))
6760, 64, 663imtr4d 286 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ → ∅ ∈ 𝐴))
68 ne0i 4180 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐵𝐵 ≠ ∅)
6967, 68impel 498 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑦𝐵) → ∅ ∈ 𝐴)
70 fconstmpt 5460 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 × {∅}) = (𝑦𝐵 ↦ ∅)
7169, 70fmptd 6699 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}):𝐵𝐴)
72 0ex 5064 . . . . . . . . . . . . . . . . . . . . . . 23 ∅ ∈ V
7372a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ∅ ∈ V)
743, 73fczfsuppd 8644 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐵 × {∅}) finSupp ∅)
7574adantr 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) finSupp ∅)
761, 2, 3cantnfs 8921 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
7776adantr 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
7871, 75, 77mpbir2and 701 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) ∈ 𝑆)
79 eqid 2771 . . . . . . . . . . . . . . . . . . 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)), ∅)
801, 33, 35, 54, 78, 79cantnfval 8923 . . . . . . . . . . . . . . . . . 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 , ∅)))
81 we0 5398 . . . . . . . . . . . . . . . . . . . . . 22 E We ∅
82 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . 23 OrdIso( E , ∅) = OrdIso( E , ∅)
8382oien 8795 . . . . . . . . . . . . . . . . . . . . . 22 ((∅ ∈ V ∧ E We ∅) → dom OrdIso( E , ∅) ≈ ∅)
8472, 81, 83mp2an 680 . . . . . . . . . . . . . . . . . . . . 21 dom OrdIso( E , ∅) ≈ ∅
85 en0 8367 . . . . . . . . . . . . . . . . . . . . 21 (dom OrdIso( E , ∅) ≈ ∅ ↔ dom OrdIso( E , ∅) = ∅)
8684, 85mpbi 222 . . . . . . . . . . . . . . . . . . . 20 dom OrdIso( E , ∅) = ∅
8786fveq2i 6499 . . . . . . . . . . . . . . . . . . 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)), ∅)‘∅)
8879seqom0g 7893 . . . . . . . . . . . . . . . . . . . 20 (∅ ∈ V → (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ∅)‘𝑘)) ·o ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +o 𝑧)), ∅)‘∅) = ∅)
8972, 88ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ∅)‘𝑘)) ·o ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +o 𝑧)), ∅)‘∅) = ∅
9087, 89eqtri 2795 . . . . . . . . . . . . . . . . . 18 (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ∅)‘𝑘)) ·o ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ∅)) = ∅
9180, 90syl6eq 2823 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
9214adantr 473 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
9392ffnd 6342 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵) Fn 𝑆)
94 fnfvelrn 6671 . . . . . . . . . . . . . . . . . 18 (((𝐴 CNF 𝐵) Fn 𝑆 ∧ (𝐵 × {∅}) ∈ 𝑆) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
9593, 78, 94syl2anc 576 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
9691, 95eqeltrrd 2860 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ∅ ∈ ran (𝐴 CNF 𝐵))
9732, 50, 96pm2.61ne 3046 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
9897expr 449 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (𝑡 ⊆ ran (𝐴 CNF 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
9931, 98sylbid 232 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
10099ex 405 . . . . . . . . . . . 12 (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
101100com23 86 . . . . . . . . . . 11 (𝜑 → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
102101a2i 14 . . . . . . . . . 10 ((𝜑 → ∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
103102a1i 11 . . . . . . . . 9 (𝑡 ∈ On → ((𝜑 → ∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))))
10423, 103syl5bi 234 . . . . . . . 8 (𝑡 ∈ On → (∀𝑦𝑡 (𝜑 → (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))))
10522, 104tfis2 7385 . . . . . . 7 (𝑡 ∈ On → (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
106105com3l 89 . . . . . 6 (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → (𝑡 ∈ On → 𝑡 ∈ ran (𝐴 CNF 𝐵))))
10718, 106mpdd 43 . . . . 5 (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
108107ssrdv 3857 . . . 4 (𝜑 → (𝐴o 𝐵) ⊆ ran (𝐴 CNF 𝐵))
10915, 108eqssd 3868 . . 3 (𝜑 → ran (𝐴 CNF 𝐵) = (𝐴o 𝐵))
110 dffo2 6420 . . 3 ((𝐴 CNF 𝐵):𝑆onto→(𝐴o 𝐵) ↔ ((𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵) ∧ ran (𝐴 CNF 𝐵) = (𝐴o 𝐵)))
11114, 109, 110sylanbrc 575 . 2 (𝜑 → (𝐴 CNF 𝐵):𝑆onto→(𝐴o 𝐵))
1122adantr 473 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝐴 ∈ On)
1133adantr 473 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝐵 ∈ On)
114 fveq2 6496 . . . . . . . . . . . 12 (𝑧 = 𝑡 → (𝑥𝑧) = (𝑥𝑡))
115 fveq2 6496 . . . . . . . . . . . 12 (𝑧 = 𝑡 → (𝑦𝑧) = (𝑦𝑡))
116114, 115eleq12d 2853 . . . . . . . . . . 11 (𝑧 = 𝑡 → ((𝑥𝑧) ∈ (𝑦𝑧) ↔ (𝑥𝑡) ∈ (𝑦𝑡)))
117 eleq1w 2841 . . . . . . . . . . . . 13 (𝑧 = 𝑡 → (𝑧𝑤𝑡𝑤))
118117imbi1d 334 . . . . . . . . . . . 12 (𝑧 = 𝑡 → ((𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
119118ralbidv 3140 . . . . . . . . . . 11 (𝑧 = 𝑡 → (∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
120116, 119anbi12d 622 . . . . . . . . . 10 (𝑧 = 𝑡 → (((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
121120cbvrexv 3377 . . . . . . . . 9 (∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
122 fveq1 6495 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (𝑥𝑡) = (𝑢𝑡))
123 fveq1 6495 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (𝑦𝑡) = (𝑣𝑡))
124 eleq12 2848 . . . . . . . . . . . 12 (((𝑥𝑡) = (𝑢𝑡) ∧ (𝑦𝑡) = (𝑣𝑡)) → ((𝑥𝑡) ∈ (𝑦𝑡) ↔ (𝑢𝑡) ∈ (𝑣𝑡)))
125122, 123, 124syl2an 587 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝑡) ∈ (𝑦𝑡) ↔ (𝑢𝑡) ∈ (𝑣𝑡)))
126 fveq1 6495 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (𝑥𝑤) = (𝑢𝑤))
127 fveq1 6495 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 → (𝑦𝑤) = (𝑣𝑤))
128126, 127eqeqan12d 2787 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝑢𝑤) = (𝑣𝑤)))
129128imbi2d 333 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤))))
130129ralbidv 3140 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → (∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤))))
131125, 130anbi12d 622 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑣) → (((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
132131rexbidv 3235 . . . . . . . . 9 ((𝑥 = 𝑢𝑦 = 𝑣) → (∃𝑡𝐵 ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
133121, 132syl5bb 275 . . . . . . . 8 ((𝑥 = 𝑢𝑦 = 𝑣) → (∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
134133cbvopabv 4997 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))}
1354, 134eqtri 2795 . . . . . 6 𝑇 = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))}
136 simprll 767 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑆)
137 simprlr 768 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑔𝑆)
138 simprr 761 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑇𝑔)
139 eqid 2771 . . . . . 6 {𝑐𝐵 ∣ (𝑓𝑐) ∈ (𝑔𝑐)} = {𝑐𝐵 ∣ (𝑓𝑐) ∈ (𝑔𝑐)}
140 eqid 2771 . . . . . 6 OrdIso( E , (𝑔 supp ∅)) = OrdIso( E , (𝑔 supp ∅))
141 eqid 2771 . . . . . 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)), ∅)
1421, 112, 113, 135, 136, 137, 138, 139, 140, 141cantnflem1 8944 . . . . 5 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
143 fvex 6509 . . . . . 6 ((𝐴 CNF 𝐵)‘𝑔) ∈ V
144143epeli 5316 . . . . 5 (((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔) ↔ ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
145142, 144sylibr 226 . . . 4 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔))
146145expr 449 . . 3 ((𝜑 ∧ (𝑓𝑆𝑔𝑆)) → (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
147146ralrimivva 3134 . 2 (𝜑 → ∀𝑓𝑆𝑔𝑆 (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
148 soisoi 6902 . 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)))
1495, 13, 111, 147, 148syl22anc 827 1 (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  wne 2960  wral 3081  wrex 3082  {crab 3085  Vcvv 3408  wss 3822  c0 4172  {csn 4435  cop 4441   cuni 4708   cint 4745   class class class wbr 4925  {copab 4987   E cep 5312   Po wpo 5320   Or wor 5321   We wwe 5361   × cxp 5401  dom cdm 5403  ran crn 5404  Ord word 6025  Oncon0 6026  cio 6147   Fn wfn 6180  wf 6181  ontowfo 6183  cfv 6185   Isom wiso 6186  (class class class)co 6974  cmpo 6976  1st c1st 7497  2nd c2nd 7498   supp csupp 7631  seq𝜔cseqom 7884   +o coa 7900   ·o comu 7901  o coe 7902  cen 8301   finSupp cfsupp 8626  OrdIsocoi 8766   CNF ccnf 8916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-supp 7632  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-seqom 7885  df-1o 7903  df-2o 7904  df-oadd 7907  df-omul 7908  df-oexp 7909  df-er 8087  df-map 8206  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-fsupp 8627  df-oi 8767  df-cnf 8917
This theorem is referenced by:  oemapwe  8949  cantnffval2  8950  cantnff1o  8951
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