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Theorem cantnf 9155
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 9139, 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 9144 . 2 (𝜑𝑇 Or 𝑆)
6 oecl 8160 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
72, 3, 6syl2anc 587 . . . 4 (𝜑 → (𝐴o 𝐵) ∈ On)
8 eloni 6190 . . . 4 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
97, 8syl 17 . . 3 (𝜑 → Ord (𝐴o 𝐵))
10 ordwe 6193 . . 3 (Ord (𝐴o 𝐵) → E We (𝐴o 𝐵))
11 weso 5534 . . 3 ( E We (𝐴o 𝐵) → E Or (𝐴o 𝐵))
12 sopo 5480 . . 3 ( E Or (𝐴o 𝐵) → E Po (𝐴o 𝐵))
139, 10, 11, 124syl 19 . 2 (𝜑 → E Po (𝐴o 𝐵))
141, 2, 3cantnff 9136 . . 3 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
1514frnd 6512 . . . 4 (𝜑 → ran (𝐴 CNF 𝐵) ⊆ (𝐴o 𝐵))
16 onss 7501 . . . . . . . 8 ((𝐴o 𝐵) ∈ On → (𝐴o 𝐵) ⊆ On)
177, 16syl 17 . . . . . . 7 (𝜑 → (𝐴o 𝐵) ⊆ On)
1817sseld 3952 . . . . . 6 (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ On))
19 eleq1w 2898 . . . . . . . . . 10 (𝑡 = 𝑦 → (𝑡 ∈ (𝐴o 𝐵) ↔ 𝑦 ∈ (𝐴o 𝐵)))
20 eleq1w 2898 . . . . . . . . . 10 (𝑡 = 𝑦 → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
2119, 20imbi12d 348 . . . . . . . . 9 (𝑡 = 𝑦 → ((𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)) ↔ (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
2221imbi2d 344 . . . . . . . 8 (𝑡 = 𝑦 → ((𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)))))
23 r19.21v 3170 . . . . . . . . 9 (∀𝑦𝑡 (𝜑 → (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → ∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
24 ordelss 6196 . . . . . . . . . . . . . . . . . . 19 ((Ord (𝐴o 𝐵) ∧ 𝑡 ∈ (𝐴o 𝐵)) → 𝑡 ⊆ (𝐴o 𝐵))
259, 24sylan 583 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → 𝑡 ⊆ (𝐴o 𝐵))
2625sselda 3953 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡 ∈ (𝐴o 𝐵)) ∧ 𝑦𝑡) → 𝑦 ∈ (𝐴o 𝐵))
27 pm5.5 365 . . . . . . . . . . . . . . . . 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 3191 . . . . . . . . . . . . . . 15 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ ∀𝑦𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵)))
30 dfss3 3941 . . . . . . . . . . . . . . 15 (𝑡 ⊆ ran (𝐴 CNF 𝐵) ↔ ∀𝑦𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵))
3129, 30syl6bbr 292 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑡 ⊆ ran (𝐴 CNF 𝐵)))
32 eleq1 2903 . . . . . . . . . . . . . . . 16 (𝑡 = ∅ → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ ∅ ∈ ran (𝐴 CNF 𝐵)))
332adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐴 ∈ On)
3433adantr 484 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐴 ∈ On)
353adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐵 ∈ On)
3635adantr 484 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐵 ∈ On)
37 simplrl 776 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ (𝐴o 𝐵))
38 simplrr 777 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ⊆ ran (𝐴 CNF 𝐵))
397adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴o 𝐵) ∈ On)
40 simprl 770 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ (𝐴o 𝐵))
41 onelon 6205 . . . . . . . . . . . . . . . . . . . 20 (((𝐴o 𝐵) ∈ On ∧ 𝑡 ∈ (𝐴o 𝐵)) → 𝑡 ∈ On)
4239, 40, 41syl2anc 587 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ On)
43 on0eln0 6235 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ On → (∅ ∈ 𝑡𝑡 ≠ ∅))
4442, 43syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝑡𝑡 ≠ ∅))
4544biimpar 481 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → ∅ ∈ 𝑡)
46 eqid 2824 . . . . . . . . . . . . . . . . 17 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴o 𝑐)} = {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴o 𝑐)}
47 eqid 2824 . . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . 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 9154 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
51 fczsupp0 7857 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 × {∅}) supp ∅) = ∅
5251eqcomi 2833 . . . . . . . . . . . . . . . . . . . 20 ∅ = ((𝐵 × {∅}) supp ∅)
53 oieq2 8976 . . . . . . . . . . . . . . . . . . . 20 (∅ = ((𝐵 × {∅}) supp ∅) → OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅)))
5452, 53ax-mp 5 . . . . . . . . . . . . . . . . . . 19 OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅))
55 ne0i 4283 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 ∈ (𝐴o 𝐵) → (𝐴o 𝐵) ≠ ∅)
5655ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴o 𝐵) ≠ ∅)
57 oveq1 7158 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
5857neeq1d 3073 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = ∅ → ((𝐴o 𝐵) ≠ ∅ ↔ (∅ ↑o 𝐵) ≠ ∅))
5956, 58syl5ibcom 248 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 = ∅ → (∅ ↑o 𝐵) ≠ ∅))
6059necon2d 3037 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((∅ ↑o 𝐵) = ∅ → 𝐴 ≠ ∅))
61 on0eln0 6235 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
62 oe0m1 8144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
6361, 62bitr3d 284 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 ∈ On → (𝐵 ≠ ∅ ↔ (∅ ↑o 𝐵) = ∅))
6435, 63syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ ↔ (∅ ↑o 𝐵) = ∅))
65 on0eln0 6235 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
6633, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝐴𝐴 ≠ ∅))
6760, 64, 663imtr4d 297 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ → ∅ ∈ 𝐴))
68 ne0i 4283 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐵𝐵 ≠ ∅)
6967, 68impel 509 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑦𝐵) → ∅ ∈ 𝐴)
70 fconstmpt 5602 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 × {∅}) = (𝑦𝐵 ↦ ∅)
7169, 70fmptd 6871 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}):𝐵𝐴)
72 0ex 5198 . . . . . . . . . . . . . . . . . . . . . . 23 ∅ ∈ V
7372a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ∅ ∈ V)
743, 73fczfsuppd 8850 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐵 × {∅}) finSupp ∅)
7574adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) finSupp ∅)
761, 2, 3cantnfs 9128 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
7776adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
7871, 75, 77mpbir2and 712 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) ∈ 𝑆)
79 eqid 2824 . . . . . . . . . . . . . . . . . . 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 9130 . . . . . . . . . . . . . . . . . 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 5538 . . . . . . . . . . . . . . . . . . . . . 22 E We ∅
82 eqid 2824 . . . . . . . . . . . . . . . . . . . . . . 23 OrdIso( E , ∅) = OrdIso( E , ∅)
8382oien 9001 . . . . . . . . . . . . . . . . . . . . . 22 ((∅ ∈ V ∧ E We ∅) → dom OrdIso( E , ∅) ≈ ∅)
8472, 81, 83mp2an 691 . . . . . . . . . . . . . . . . . . . . 21 dom OrdIso( E , ∅) ≈ ∅
85 en0 8570 . . . . . . . . . . . . . . . . . . . . 21 (dom OrdIso( E , ∅) ≈ ∅ ↔ dom OrdIso( E , ∅) = ∅)
8684, 85mpbi 233 . . . . . . . . . . . . . . . . . . . 20 dom OrdIso( E , ∅) = ∅
8786fveq2i 6666 . . . . . . . . . . . . . . . . . . 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 8090 . . . . . . . . . . . . . . . . . . . 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 2847 . . . . . . . . . . . . . . . . . 18 (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ∅)‘𝑘)) ·o ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ∅)) = ∅
9180, 90syl6eq 2875 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
9214adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
9392ffnd 6506 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵) Fn 𝑆)
94 fnfvelrn 6841 . . . . . . . . . . . . . . . . . 18 (((𝐴 CNF 𝐵) Fn 𝑆 ∧ (𝐵 × {∅}) ∈ 𝑆) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
9593, 78, 94syl2anc 587 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
9691, 95eqeltrrd 2917 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ∅ ∈ ran (𝐴 CNF 𝐵))
9732, 50, 96pm2.61ne 3099 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
9897expr 460 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (𝑡 ⊆ ran (𝐴 CNF 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
9931, 98sylbid 243 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
10099ex 416 . . . . . . . . . . . 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 245 . . . . . . . 8 (𝑡 ∈ On → (∀𝑦𝑡 (𝜑 → (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))))
10522, 104tfis2 7567 . . . . . . 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 3959 . . . 4 (𝜑 → (𝐴o 𝐵) ⊆ ran (𝐴 CNF 𝐵))
10915, 108eqssd 3970 . . 3 (𝜑 → ran (𝐴 CNF 𝐵) = (𝐴o 𝐵))
110 dffo2 6587 . . 3 ((𝐴 CNF 𝐵):𝑆onto→(𝐴o 𝐵) ↔ ((𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵) ∧ ran (𝐴 CNF 𝐵) = (𝐴o 𝐵)))
11114, 109, 110sylanbrc 586 . 2 (𝜑 → (𝐴 CNF 𝐵):𝑆onto→(𝐴o 𝐵))
1122adantr 484 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝐴 ∈ On)
1133adantr 484 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝐵 ∈ On)
114 fveq2 6663 . . . . . . . . . . . 12 (𝑧 = 𝑡 → (𝑥𝑧) = (𝑥𝑡))
115 fveq2 6663 . . . . . . . . . . . 12 (𝑧 = 𝑡 → (𝑦𝑧) = (𝑦𝑡))
116114, 115eleq12d 2910 . . . . . . . . . . 11 (𝑧 = 𝑡 → ((𝑥𝑧) ∈ (𝑦𝑧) ↔ (𝑥𝑡) ∈ (𝑦𝑡)))
117 eleq1w 2898 . . . . . . . . . . . . 13 (𝑧 = 𝑡 → (𝑧𝑤𝑡𝑤))
118117imbi1d 345 . . . . . . . . . . . 12 (𝑧 = 𝑡 → ((𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
119118ralbidv 3192 . . . . . . . . . . 11 (𝑧 = 𝑡 → (∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
120116, 119anbi12d 633 . . . . . . . . . 10 (𝑧 = 𝑡 → (((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
121120cbvrexvw 3435 . . . . . . . . 9 (∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
122 fveq1 6662 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (𝑥𝑡) = (𝑢𝑡))
123 fveq1 6662 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (𝑦𝑡) = (𝑣𝑡))
124 eleq12 2905 . . . . . . . . . . . 12 (((𝑥𝑡) = (𝑢𝑡) ∧ (𝑦𝑡) = (𝑣𝑡)) → ((𝑥𝑡) ∈ (𝑦𝑡) ↔ (𝑢𝑡) ∈ (𝑣𝑡)))
125122, 123, 124syl2an 598 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝑡) ∈ (𝑦𝑡) ↔ (𝑢𝑡) ∈ (𝑣𝑡)))
126 fveq1 6662 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (𝑥𝑤) = (𝑢𝑤))
127 fveq1 6662 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 → (𝑦𝑤) = (𝑣𝑤))
128126, 127eqeqan12d 2841 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝑢𝑤) = (𝑣𝑤)))
129128imbi2d 344 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤))))
130129ralbidv 3192 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → (∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤))))
131125, 130anbi12d 633 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑣) → (((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
132131rexbidv 3289 . . . . . . . . 9 ((𝑥 = 𝑢𝑦 = 𝑣) → (∃𝑡𝐵 ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
133121, 132syl5bb 286 . . . . . . . 8 ((𝑥 = 𝑢𝑦 = 𝑣) → (∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
134133cbvopabv 5125 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))}
1354, 134eqtri 2847 . . . . . 6 𝑇 = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))}
136 simprll 778 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑆)
137 simprlr 779 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑔𝑆)
138 simprr 772 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑇𝑔)
139 eqid 2824 . . . . . 6 {𝑐𝐵 ∣ (𝑓𝑐) ∈ (𝑔𝑐)} = {𝑐𝐵 ∣ (𝑓𝑐) ∈ (𝑔𝑐)}
140 eqid 2824 . . . . . 6 OrdIso( E , (𝑔 supp ∅)) = OrdIso( E , (𝑔 supp ∅))
141 eqid 2824 . . . . . 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 9151 . . . . 5 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
143 fvex 6676 . . . . . 6 ((𝐴 CNF 𝐵)‘𝑔) ∈ V
144143epeli 5456 . . . . 5 (((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔) ↔ ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
145142, 144sylibr 237 . . . 4 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔))
146145expr 460 . . 3 ((𝜑 ∧ (𝑓𝑆𝑔𝑆)) → (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
147146ralrimivva 3186 . 2 (𝜑 → ∀𝑓𝑆𝑔𝑆 (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
148 soisoi 7076 . 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 837 1 (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  {crab 3137  Vcvv 3480  wss 3919  c0 4276  {csn 4550  cop 4556   cuni 4824   cint 4862   class class class wbr 5053  {copab 5115   E cep 5452   Po wpo 5460   Or wor 5461   We wwe 5501   × cxp 5541  dom cdm 5543  ran crn 5544  Ord word 6179  Oncon0 6180  cio 6302   Fn wfn 6340  wf 6341  ontowfo 6343  cfv 6345   Isom wiso 6346  (class class class)co 7151  cmpo 7153  1st c1st 7684  2nd c2nd 7685   supp csupp 7828  seqωcseqom 8081   +o coa 8097   ·o comu 8098  o coe 8099  cen 8504   finSupp cfsupp 8832  OrdIsocoi 8972   CNF ccnf 9123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-isom 6354  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-supp 7829  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-seqom 8082  df-1o 8100  df-2o 8101  df-oadd 8104  df-omul 8105  df-oexp 8106  df-er 8287  df-map 8406  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-fsupp 8833  df-oi 8973  df-cnf 9124
This theorem is referenced by:  oemapwe  9156  cantnffval2  9157  cantnff1o  9158
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