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Theorem oeicl 6070
 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 6064 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
2 vex 2575 . . . . . 6 𝑥 ∈ V
3 omexg 6059 . . . . . 6 ((𝑥 ∈ V ∧ 𝐴 ∈ On) → (𝑥 ·𝑜 𝐴) ∈ V)
42, 3mpan 408 . . . . 5 (𝐴 ∈ On → (𝑥 ·𝑜 𝐴) ∈ V)
54ralrimivw 2408 . . . 4 (𝐴 ∈ On → ∀𝑥 ∈ V (𝑥 ·𝑜 𝐴) ∈ V)
6 eqid 2054 . . . . 5 (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))
76fnmpt 5050 . . . 4 (∀𝑥 ∈ V (𝑥 ·𝑜 𝐴) ∈ V → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) Fn V)
85, 7syl 14 . . 3 (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) Fn V)
9 1on 6036 . . . 4 1𝑜 ∈ On
109a1i 9 . . 3 (𝐴 ∈ On → 1𝑜 ∈ On)
11 omcl 6069 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ·𝑜 𝐴) ∈ On)
12 vex 2575 . . . . . . . 8 𝑦 ∈ V
13 oveq1 5544 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ·𝑜 𝐴) = (𝑦 ·𝑜 𝐴))
1413, 6fvmptg 5273 . . . . . . . 8 ((𝑦 ∈ V ∧ (𝑦 ·𝑜 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) = (𝑦 ·𝑜 𝐴))
1512, 14mpan 408 . . . . . . 7 ((𝑦 ·𝑜 𝐴) ∈ On → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) = (𝑦 ·𝑜 𝐴))
1611, 15syl 14 . . . . . 6 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) = (𝑦 ·𝑜 𝐴))
1716, 11eqeltrd 2128 . . . . 5 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) ∈ On)
1817ancoms 259 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) ∈ On)
1918ralrimiva 2407 . . 3 (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) ∈ On)
208, 10, 19rdgon 6001 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ On)
211, 20eqeltrd 2128 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1257   ∈ wcel 1407  ∀wral 2321  Vcvv 2572   ↦ cmpt 3843  Oncon0 4125   Fn wfn 4922  ‘cfv 4927  (class class class)co 5537  reccrdg 5984  1𝑜c1o 6022   ·𝑜 comu 6027   ↑𝑜 coei 6028 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 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287 This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-id 4055  df-iord 4128  df-on 4130  df-suc 4133  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-1o 6029  df-oadd 6033  df-omul 6034  df-oexpi 6035 This theorem is referenced by: (None)
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