Theorem oemapwe 9157

Description: The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternate definition of the ordinal exponential. (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
oemapwe	⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 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 oemapwe

Step	Hyp	Ref	Expression
1		cantnfs.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ On)
2		cantnfs.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ On)
3		oecl 8162	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
4	1, 2, 3	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On)
5		eloni 6201	. . . 4 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵))
6		ordwe 6204	. . . 4 ⊢ (Ord (𝐴 ↑o 𝐵) → E We (𝐴 ↑o 𝐵))
7	4, 5, 6	3syl 18	. . 3 ⊢ (𝜑 → E We (𝐴 ↑o 𝐵))
8		cantnfs.s	. . . . 5 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
9		oemapval.t	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
10	8, 1, 2, 9	cantnf 9156	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)))
11		isowe 7102	. . . 4 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)) → (𝑇 We 𝑆 ↔ E We (𝐴 ↑o 𝐵)))
12	10, 11	syl 17	. . 3 ⊢ (𝜑 → (𝑇 We 𝑆 ↔ E We (𝐴 ↑o 𝐵)))
13	7, 12	mpbird 259	. 2 ⊢ (𝜑 → 𝑇 We 𝑆)
14	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → Ord (𝐴 ↑o 𝐵))
15		isocnv 7083	. . . . . 6 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)) → ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆))
16	10, 15	syl 17	. . . . 5 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆))
17		ovex 7189	. . . . . . . . 9 ⊢ (𝐴 CNF 𝐵) ∈ V
18	17	dmex 7616	. . . . . . . 8 ⊢ dom (𝐴 CNF 𝐵) ∈ V
19	8, 18	eqeltri 2909	. . . . . . 7 ⊢ 𝑆 ∈ V
20		exse 5519	. . . . . . 7 ⊢ (𝑆 ∈ V → 𝑇 Se 𝑆)
21	19, 20	ax-mp 5	. . . . . 6 ⊢ 𝑇 Se 𝑆
22		eqid 2821	. . . . . . 7 ⊢ OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆)
23	22	oieu 9003	. . . . . 6 ⊢ ((𝑇 We 𝑆 ∧ 𝑇 Se 𝑆) → ((Ord (𝐴 ↑o 𝐵) ∧ ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) ↔ ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))))
24	13, 21, 23	sylancl 588	. . . . 5 ⊢ (𝜑 → ((Ord (𝐴 ↑o 𝐵) ∧ ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) ↔ ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))))
25	14, 16, 24	mpbi2and 710	. . . 4 ⊢ (𝜑 → ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))
26	25	simpld 497	. . 3 ⊢ (𝜑 → (𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆))
27	26	eqcomd 2827	. 2 ⊢ (𝜑 → dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵))
28	13, 27	jca 514	1 ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  Vcvv 3494  {copab 5128   E cep 5464   Se wse 5512   We wwe 5513  ^◡ccnv 5554  dom cdm 5555  Ord word 6190  Oncon0 6191  ‘cfv 6355   Isom wiso 6356  (class class class)co 7156   ↑_o coe 8101  OrdIsocoi 8973   CNF ccnf 9124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-seqom 8084  df-1o 8102  df-2o 8103  df-oadd 8106  df-omul 8107  df-oexp 8108  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-oi 8974  df-cnf 9125

This theorem is referenced by:  cantnffval2  9158  wemapwe  9160