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Theorem oeicl 6456
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 6450 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
2 1on 6417 . . . 4 1o ∈ On
32a1i 9 . . 3 (𝐴 ∈ On → 1o ∈ On)
4 vex 2740 . . . . . . 7 𝑦 ∈ V
5 omcl 6455 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ·o 𝐴) ∈ On)
6 oveq1 5875 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 ·o 𝐴) = (𝑦 ·o 𝐴))
7 eqid 2177 . . . . . . . 8 (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
86, 7fvmptg 5587 . . . . . . 7 ((𝑦 ∈ V ∧ (𝑦 ·o 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) = (𝑦 ·o 𝐴))
94, 5, 8sylancr 414 . . . . . 6 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) = (𝑦 ·o 𝐴))
109, 5eqeltrd 2254 . . . . 5 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
1110ancoms 268 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
1211ralrimiva 2550 . . 3 (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
133, 12rdgon 6380 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ On)
141, 13eqeltrd 2254 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  cmpt 4061  Oncon0 4359  cfv 5211  (class class class)co 5868  reccrdg 6363  1oc1o 6403   ·o comu 6408  o coei 6409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-1o 6410  df-oadd 6414  df-omul 6415  df-oexpi 6416
This theorem is referenced by: (None)
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