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Theorem oeicl 6362
 Description: Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
Assertion
Ref Expression
oeicl ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)

Proof of Theorem oeicl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oeiv 6356 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
2 1on 6324 . . . 4 1o ∈ On
32a1i 9 . . 3 (𝐴 ∈ On → 1o ∈ On)
4 vex 2690 . . . . . . 7 𝑦 ∈ V
5 omcl 6361 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ·o 𝐴) ∈ On)
6 oveq1 5785 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 ·o 𝐴) = (𝑦 ·o 𝐴))
7 eqid 2140 . . . . . . . 8 (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
86, 7fvmptg 5501 . . . . . . 7 ((𝑦 ∈ V ∧ (𝑦 ·o 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) = (𝑦 ·o 𝐴))
94, 5, 8sylancr 411 . . . . . 6 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) = (𝑦 ·o 𝐴))
109, 5eqeltrd 2217 . . . . 5 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
1110ancoms 266 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
1211ralrimiva 2506 . . 3 (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
133, 12rdgon 6287 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ On)
141, 13eqeltrd 2217 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  Vcvv 2687   ↦ cmpt 3993  Oncon0 4289  ‘cfv 5127  (class class class)co 5778  reccrdg 6270  1oc1o 6310   ·o comu 6315   ↑o coei 6316 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-iord 4292  df-on 4294  df-suc 4297  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-1o 6317  df-oadd 6321  df-omul 6322  df-oexpi 6323 This theorem is referenced by: (None)
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