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Theorem oeiv 6066
 Description: Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)
Assertion
Ref Expression
oeiv ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oeiv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1on 6038 . . 3 1𝑜 ∈ On
2 vex 2577 . . . . . . 7 𝑥 ∈ V
3 omexg 6061 . . . . . . 7 ((𝑥 ∈ V ∧ 𝐴 ∈ On) → (𝑥 ·𝑜 𝐴) ∈ V)
42, 3mpan 408 . . . . . 6 (𝐴 ∈ On → (𝑥 ·𝑜 𝐴) ∈ V)
54ralrimivw 2410 . . . . 5 (𝐴 ∈ On → ∀𝑥 ∈ V (𝑥 ·𝑜 𝐴) ∈ V)
6 eqid 2056 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))
76fnmpt 5052 . . . . 5 (∀𝑥 ∈ V (𝑥 ·𝑜 𝐴) ∈ V → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) Fn V)
85, 7syl 14 . . . 4 (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) Fn V)
9 rdgexggg 5994 . . . 4 (((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) Fn V ∧ 1𝑜 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V)
108, 9syl3an1 1179 . . 3 ((𝐴 ∈ On ∧ 1𝑜 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V)
111, 10mp3an2 1231 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V)
12 oveq2 5547 . . . . . 6 (𝑦 = 𝐴 → (𝑥 ·𝑜 𝑦) = (𝑥 ·𝑜 𝐴))
1312mpteq2dv 3875 . . . . 5 (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)))
14 rdgeq1 5988 . . . . 5 ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜) = rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜))
1513, 14syl 14 . . . 4 (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜) = rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜))
1615fveq1d 5207 . . 3 (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧))
17 fveq2 5205 . . 3 (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
18 df-oexpi 6037 . . 3 𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧))
1916, 17, 18ovmpt2g 5662 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
2011, 19mpd3an3 1244 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259   ∈ wcel 1409  ∀wral 2323  Vcvv 2574   ↦ cmpt 3845  Oncon0 4127   Fn wfn 4924  ‘cfv 4929  (class class class)co 5539  reccrdg 5986  1𝑜c1o 6024   ·𝑜 comu 6029   ↑𝑜 coei 6030 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-oexpi 6037 This theorem is referenced by:  oei0  6069  oeicl  6072
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