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

Theorem cantnf 9731
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 9715, 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 9720 . 2 (𝜑𝑇 Or 𝑆)
6 oecl 8574 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
72, 3, 6syl2anc 584 . . . 4 (𝜑 → (𝐴o 𝐵) ∈ On)
8 eloni 6396 . . . 4 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
97, 8syl 17 . . 3 (𝜑 → Ord (𝐴o 𝐵))
10 ordwe 6399 . . 3 (Ord (𝐴o 𝐵) → E We (𝐴o 𝐵))
11 weso 5680 . . 3 ( E We (𝐴o 𝐵) → E Or (𝐴o 𝐵))
12 sopo 5616 . . 3 ( E Or (𝐴o 𝐵) → E Po (𝐴o 𝐵))
139, 10, 11, 124syl 19 . 2 (𝜑 → E Po (𝐴o 𝐵))
141, 2, 3cantnff 9712 . . 3 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
1514frnd 6745 . . . 4 (𝜑 → ran (𝐴 CNF 𝐵) ⊆ (𝐴o 𝐵))
16 onss 7804 . . . . . . . 8 ((𝐴o 𝐵) ∈ On → (𝐴o 𝐵) ⊆ On)
177, 16syl 17 . . . . . . 7 (𝜑 → (𝐴o 𝐵) ⊆ On)
1817sseld 3994 . . . . . 6 (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ On))
19 eleq1w 2822 . . . . . . . . . 10 (𝑡 = 𝑦 → (𝑡 ∈ (𝐴o 𝐵) ↔ 𝑦 ∈ (𝐴o 𝐵)))
20 eleq1w 2822 . . . . . . . . . 10 (𝑡 = 𝑦 → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ 𝑦 ∈ ran (𝐴 CNF 𝐵)))
2119, 20imbi12d 344 . . . . . . . . 9 (𝑡 = 𝑦 → ((𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)) ↔ (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
2221imbi2d 340 . . . . . . . 8 (𝑡 = 𝑦 → ((𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)))))
23 r19.21v 3178 . . . . . . . . 9 (∀𝑦𝑡 (𝜑 → (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) ↔ (𝜑 → ∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))))
24 ordelss 6402 . . . . . . . . . . . . . . . . . . 19 ((Ord (𝐴o 𝐵) ∧ 𝑡 ∈ (𝐴o 𝐵)) → 𝑡 ⊆ (𝐴o 𝐵))
259, 24sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → 𝑡 ⊆ (𝐴o 𝐵))
2625sselda 3995 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡 ∈ (𝐴o 𝐵)) ∧ 𝑦𝑡) → 𝑦 ∈ (𝐴o 𝐵))
27 pm5.5 361 . . . . . . . . . . . . . . . . 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 3174 . . . . . . . . . . . . . . 15 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ ∀𝑦𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵)))
30 dfss3 3984 . . . . . . . . . . . . . . 15 (𝑡 ⊆ ran (𝐴 CNF 𝐵) ↔ ∀𝑦𝑡 𝑦 ∈ ran (𝐴 CNF 𝐵))
3129, 30bitr4di 289 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) ↔ 𝑡 ⊆ ran (𝐴 CNF 𝐵)))
32 eleq1 2827 . . . . . . . . . . . . . . . 16 (𝑡 = ∅ → (𝑡 ∈ ran (𝐴 CNF 𝐵) ↔ ∅ ∈ ran (𝐴 CNF 𝐵)))
332adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐴 ∈ On)
3433adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐴 ∈ On)
353adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝐵 ∈ On)
3635adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝐵 ∈ On)
37 simplrl 777 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ (𝐴o 𝐵))
38 simplrr 778 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ⊆ ran (𝐴 CNF 𝐵))
397adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴o 𝐵) ∈ On)
40 simprl 771 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ (𝐴o 𝐵))
41 onelon 6411 . . . . . . . . . . . . . . . . . . . 20 (((𝐴o 𝐵) ∈ On ∧ 𝑡 ∈ (𝐴o 𝐵)) → 𝑡 ∈ On)
4239, 40, 41syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ On)
43 on0eln0 6442 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ On → (∅ ∈ 𝑡𝑡 ≠ ∅))
4442, 43syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝑡𝑡 ≠ ∅))
4544biimpar 477 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → ∅ ∈ 𝑡)
46 eqid 2735 . . . . . . . . . . . . . . . . 17 {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴o 𝑐)} = {𝑐 ∈ On ∣ 𝑡 ∈ (𝐴o 𝑐)}
47 eqid 2735 . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 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 9730 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑡 ≠ ∅) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
51 fczsupp0 8217 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 × {∅}) supp ∅) = ∅
5251eqcomi 2744 . . . . . . . . . . . . . . . . . . . 20 ∅ = ((𝐵 × {∅}) supp ∅)
53 oieq2 9551 . . . . . . . . . . . . . . . . . . . 20 (∅ = ((𝐵 × {∅}) supp ∅) → OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅)))
5452, 53ax-mp 5 . . . . . . . . . . . . . . . . . . 19 OrdIso( E , ∅) = OrdIso( E , ((𝐵 × {∅}) supp ∅))
55 ne0i 4347 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 ∈ (𝐴o 𝐵) → (𝐴o 𝐵) ≠ ∅)
5655ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴o 𝐵) ≠ ∅)
57 oveq1 7438 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
5857neeq1d 2998 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = ∅ → ((𝐴o 𝐵) ≠ ∅ ↔ (∅ ↑o 𝐵) ≠ ∅))
5956, 58syl5ibcom 245 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 = ∅ → (∅ ↑o 𝐵) ≠ ∅))
6059necon2d 2961 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((∅ ↑o 𝐵) = ∅ → 𝐴 ≠ ∅))
61 on0eln0 6442 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
62 oe0m1 8558 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
6361, 62bitr3d 281 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 ∈ On → (𝐵 ≠ ∅ ↔ (∅ ↑o 𝐵) = ∅))
6435, 63syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ ↔ (∅ ↑o 𝐵) = ∅))
65 on0eln0 6442 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
6633, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (∅ ∈ 𝐴𝐴 ≠ ∅))
6760, 64, 663imtr4d 294 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 ≠ ∅ → ∅ ∈ 𝐴))
68 ne0i 4347 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐵𝐵 ≠ ∅)
6967, 68impel 505 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) ∧ 𝑦𝐵) → ∅ ∈ 𝐴)
70 fconstmpt 5751 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 × {∅}) = (𝑦𝐵 ↦ ∅)
7169, 70fmptd 7134 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}):𝐵𝐴)
72 0ex 5313 . . . . . . . . . . . . . . . . . . . . . . 23 ∅ ∈ V
7372a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ∅ ∈ V)
743, 73fczfsuppd 9424 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐵 × {∅}) finSupp ∅)
7574adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) finSupp ∅)
761, 2, 3cantnfs 9704 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
7776adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
7871, 75, 77mpbir2and 713 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐵 × {∅}) ∈ 𝑆)
79 eqid 2735 . . . . . . . . . . . . . . . . . . 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 9706 . . . . . . . . . . . . . . . . . 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 5684 . . . . . . . . . . . . . . . . . . . . . 22 E We ∅
82 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . 23 OrdIso( E , ∅) = OrdIso( E , ∅)
8382oien 9576 . . . . . . . . . . . . . . . . . . . . . 22 ((∅ ∈ V ∧ E We ∅) → dom OrdIso( E , ∅) ≈ ∅)
8472, 81, 83mp2an 692 . . . . . . . . . . . . . . . . . . . . 21 dom OrdIso( E , ∅) ≈ ∅
85 en0 9057 . . . . . . . . . . . . . . . . . . . . 21 (dom OrdIso( E , ∅) ≈ ∅ ↔ dom OrdIso( E , ∅) = ∅)
8684, 85mpbi 230 . . . . . . . . . . . . . . . . . . . 20 dom OrdIso( E , ∅) = ∅
8786fveq2i 6910 . . . . . . . . . . . . . . . . . . 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 8495 . . . . . . . . . . . . . . . . . . . 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 2763 . . . . . . . . . . . . . . . . . 18 (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ∅)‘𝑘)) ·o ((𝐵 × {∅})‘(OrdIso( E , ∅)‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ∅)) = ∅
9180, 90eqtrdi 2791 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
9214adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
9392ffnd 6738 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → (𝐴 CNF 𝐵) Fn 𝑆)
94 fnfvelrn 7100 . . . . . . . . . . . . . . . . . 18 (((𝐴 CNF 𝐵) Fn 𝑆 ∧ (𝐵 × {∅}) ∈ 𝑆) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
9593, 78, 94syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) ∈ ran (𝐴 CNF 𝐵))
9691, 95eqeltrrd 2840 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → ∅ ∈ ran (𝐴 CNF 𝐵))
9732, 50, 96pm2.61ne 3025 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡 ∈ (𝐴o 𝐵) ∧ 𝑡 ⊆ ran (𝐴 CNF 𝐵))) → 𝑡 ∈ ran (𝐴 CNF 𝐵))
9897expr 456 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (𝑡 ⊆ ran (𝐴 CNF 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
9931, 98sylbid 240 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ (𝐴o 𝐵)) → (∀𝑦𝑡 (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵)) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))
10099ex 412 . . . . . . . . . . . 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, 103biimtrid 242 . . . . . . . 8 (𝑡 ∈ On → (∀𝑦𝑡 (𝜑 → (𝑦 ∈ (𝐴o 𝐵) → 𝑦 ∈ ran (𝐴 CNF 𝐵))) → (𝜑 → (𝑡 ∈ (𝐴o 𝐵) → 𝑡 ∈ ran (𝐴 CNF 𝐵)))))
10522, 104tfis2 7878 . . . . . . 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 4001 . . . 4 (𝜑 → (𝐴o 𝐵) ⊆ ran (𝐴 CNF 𝐵))
10915, 108eqssd 4013 . . 3 (𝜑 → ran (𝐴 CNF 𝐵) = (𝐴o 𝐵))
110 dffo2 6825 . . 3 ((𝐴 CNF 𝐵):𝑆onto→(𝐴o 𝐵) ↔ ((𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵) ∧ ran (𝐴 CNF 𝐵) = (𝐴o 𝐵)))
11114, 109, 110sylanbrc 583 . 2 (𝜑 → (𝐴 CNF 𝐵):𝑆onto→(𝐴o 𝐵))
1122adantr 480 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝐴 ∈ On)
1133adantr 480 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝐵 ∈ On)
114 fveq2 6907 . . . . . . . . . . . 12 (𝑧 = 𝑡 → (𝑥𝑧) = (𝑥𝑡))
115 fveq2 6907 . . . . . . . . . . . 12 (𝑧 = 𝑡 → (𝑦𝑧) = (𝑦𝑡))
116114, 115eleq12d 2833 . . . . . . . . . . 11 (𝑧 = 𝑡 → ((𝑥𝑧) ∈ (𝑦𝑧) ↔ (𝑥𝑡) ∈ (𝑦𝑡)))
117 eleq1w 2822 . . . . . . . . . . . . 13 (𝑧 = 𝑡 → (𝑧𝑤𝑡𝑤))
118117imbi1d 341 . . . . . . . . . . . 12 (𝑧 = 𝑡 → ((𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
119118ralbidv 3176 . . . . . . . . . . 11 (𝑧 = 𝑡 → (∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
120116, 119anbi12d 632 . . . . . . . . . 10 (𝑧 = 𝑡 → (((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
121120cbvrexvw 3236 . . . . . . . . 9 (∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))))
122 fveq1 6906 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (𝑥𝑡) = (𝑢𝑡))
123 fveq1 6906 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (𝑦𝑡) = (𝑣𝑡))
124 eleq12 2829 . . . . . . . . . . . 12 (((𝑥𝑡) = (𝑢𝑡) ∧ (𝑦𝑡) = (𝑣𝑡)) → ((𝑥𝑡) ∈ (𝑦𝑡) ↔ (𝑢𝑡) ∈ (𝑣𝑡)))
125122, 123, 124syl2an 596 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝑡) ∈ (𝑦𝑡) ↔ (𝑢𝑡) ∈ (𝑣𝑡)))
126 fveq1 6906 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (𝑥𝑤) = (𝑢𝑤))
127 fveq1 6906 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 → (𝑦𝑤) = (𝑣𝑤))
128126, 127eqeqan12d 2749 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝑢𝑤) = (𝑣𝑤)))
129128imbi2d 340 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤))))
130129ralbidv 3176 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → (∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤))))
131125, 130anbi12d 632 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑣) → (((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
132131rexbidv 3177 . . . . . . . . 9 ((𝑥 = 𝑢𝑦 = 𝑣) → (∃𝑡𝐵 ((𝑥𝑡) ∈ (𝑦𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
133121, 132bitrid 283 . . . . . . . 8 ((𝑥 = 𝑢𝑦 = 𝑣) → (∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))))
134133cbvopabv 5221 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))}
1354, 134eqtri 2763 . . . . . 6 𝑇 = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡𝐵 ((𝑢𝑡) ∈ (𝑣𝑡) ∧ ∀𝑤𝐵 (𝑡𝑤 → (𝑢𝑤) = (𝑣𝑤)))}
136 simprll 779 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑆)
137 simprlr 780 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑔𝑆)
138 simprr 773 . . . . . 6 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → 𝑓𝑇𝑔)
139 eqid 2735 . . . . . 6 {𝑐𝐵 ∣ (𝑓𝑐) ∈ (𝑔𝑐)} = {𝑐𝐵 ∣ (𝑓𝑐) ∈ (𝑔𝑐)}
140 eqid 2735 . . . . . 6 OrdIso( E , (𝑔 supp ∅)) = OrdIso( E , (𝑔 supp ∅))
141 eqid 2735 . . . . . 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 9727 . . . . 5 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
143 fvex 6920 . . . . . 6 ((𝐴 CNF 𝐵)‘𝑔) ∈ V
144143epeli 5591 . . . . 5 (((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔) ↔ ((𝐴 CNF 𝐵)‘𝑓) ∈ ((𝐴 CNF 𝐵)‘𝑔))
145142, 144sylibr 234 . . . 4 ((𝜑 ∧ ((𝑓𝑆𝑔𝑆) ∧ 𝑓𝑇𝑔)) → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔))
146145expr 456 . . 3 ((𝜑 ∧ (𝑓𝑆𝑔𝑆)) → (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
147146ralrimivva 3200 . 2 (𝜑 → ∀𝑓𝑆𝑔𝑆 (𝑓𝑇𝑔 → ((𝐴 CNF 𝐵)‘𝑓) E ((𝐴 CNF 𝐵)‘𝑔)))
148 soisoi 7348 . 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 839 1 (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339  {csn 4631  cop 4637   cuni 4912   cint 4951   class class class wbr 5148  {copab 5210   E cep 5588   Po wpo 5595   Or wor 5596   We wwe 5640   × cxp 5687  dom cdm 5689  ran crn 5690  Ord word 6385  Oncon0 6386  cio 6514   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563   Isom wiso 6564  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012   supp csupp 8184  seqωcseqom 8486   +o coa 8502   ·o comu 8503  o coe 8504  cen 8981   finSupp cfsupp 9399  OrdIsocoi 9547   CNF ccnf 9699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-cnf 9700
This theorem is referenced by:  oemapwe  9732  cantnffval2  9733  cantnff1o  9734  cantnfresb  43314
  Copyright terms: Public domain W3C validator