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Theorem fnoei 6032
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 6007 . 2 𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦))
2 vex 2560 . . 3 𝑦 ∈ V
3 1on 6008 . . . . 5 1𝑜 ∈ On
43elexi 2567 . . . 4 1𝑜 ∈ V
5 vex 2560 . . . . . 6 𝑧 ∈ V
6 vex 2560 . . . . . 6 𝑥 ∈ V
7 omexg 6031 . . . . . 6 ((𝑧 ∈ V ∧ 𝑥 ∈ V) → (𝑧 ·𝑜 𝑥) ∈ V)
85, 6, 7mp2an 402 . . . . 5 (𝑧 ·𝑜 𝑥) ∈ V
9 eqid 2040 . . . . 5 (𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)) = (𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥))
108, 9fnmpti 5027 . . . 4 (𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)) Fn V
114, 10rdgexg 5976 . . 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 5830 1 𝑜 Fn (On × On)
Colors of variables: wff set class
Syntax hints:  wcel 1393  Vcvv 2557  cmpt 3818  Oncon0 4100   × cxp 4343   Fn wfn 4897  cfv 4902  (class class class)co 5512  reccrdg 5956  1𝑜c1o 5994   ·𝑜 comu 5999  𝑜 coei 6000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-oexpi 6007
This theorem is referenced by: (None)
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