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Theorem fnoei 6062
Description: Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
Assertion
Ref Expression
fnoei 𝑜 Fn (On × On)

Proof of Theorem fnoei
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-oexpi 6037 . 2 𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦))
2 vex 2577 . . 3 𝑦 ∈ V
3 1on 6038 . . . . 5 1𝑜 ∈ On
43elexi 2584 . . . 4 1𝑜 ∈ V
5 vex 2577 . . . . . 6 𝑧 ∈ V
6 vex 2577 . . . . . 6 𝑥 ∈ V
7 omexg 6061 . . . . . 6 ((𝑧 ∈ V ∧ 𝑥 ∈ V) → (𝑧 ·𝑜 𝑥) ∈ V)
85, 6, 7mp2an 410 . . . . 5 (𝑧 ·𝑜 𝑥) ∈ V
9 eqid 2056 . . . . 5 (𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)) = (𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥))
108, 9fnmpti 5054 . . . 4 (𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)) Fn V
114, 10rdgexg 6006 . . 3 (𝑦 ∈ V → (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦) ∈ V)
122, 11ax-mp 7 . 2 (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦) ∈ V
131, 12fnmpt2i 5857 1 𝑜 Fn (On × On)
Colors of variables: wff set class
Syntax hints:  wcel 1409  Vcvv 2574  cmpt 3845  Oncon0 4127   × cxp 4370   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: (None)
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