Theorem oemapwe 8591

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 7617	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
4	1, 2, 3	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐴 ↑𝑜 𝐵) ∈ On)
5		eloni 5733	. . . 4 ⊢ ((𝐴 ↑𝑜 𝐵) ∈ On → Ord (𝐴 ↑𝑜 𝐵))
6		ordwe 5736	. . . 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 8590	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑𝑜 𝐵)))
11		isowe 6599	. . . 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 247	. 2 ⊢ (𝜑 → 𝑇 We 𝑆)
14	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → Ord (𝐴 ↑𝑜 𝐵))
15		isocnv 6580	. . . . . 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 6678	. . . . . . . . 9 ⊢ (𝐴 CNF 𝐵) ∈ V
18	17	dmex 7099	. . . . . . . 8 ⊢ dom (𝐴 CNF 𝐵) ∈ V
19	8, 18	eqeltri 2697	. . . . . . 7 ⊢ 𝑆 ∈ V
20		exse 5078	. . . . . . 7 ⊢ (𝑆 ∈ V → 𝑇 Se 𝑆)
21	19, 20	ax-mp 5	. . . . . 6 ⊢ 𝑇 Se 𝑆
22		eqid 2622	. . . . . . 7 ⊢ OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆)
23	22	oieu 8444	. . . . . 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 694	. . . . 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 956	. . . 4 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))
26	25	simpld 475	. . 3 ⊢ (𝜑 → (𝐴 ↑𝑜 𝐵) = dom OrdIso(𝑇, 𝑆))
27	26	eqcomd 2628	. 2 ⊢ (𝜑 → dom OrdIso(𝑇, 𝑆) = (𝐴 ↑𝑜 𝐵))
28	13, 27	jca 554	1 ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑𝑜 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200  {copab 4712   E cep 5028   Se wse 5071   We wwe 5072  ^◡ccnv 5113  dom cdm 5114  Ord word 5722  Oncon0 5723  ‘cfv 5888   Isom wiso 5889  (class class class)co 6650   ↑_𝑜 coe 7559  OrdIsocoi 8414   CNF ccnf 8558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559

This theorem is referenced by:  cantnffval2  8592  wemapwe  8594