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Theorem fnoe 8124
Description: Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
fnoe o Fn (On × On)

Proof of Theorem fnoe
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-oexp 8097 . 2 o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
2 1on 8098 . . . 4 1o ∈ On
3 difexg 5222 . . . 4 (1o ∈ On → (1o𝑦) ∈ V)
42, 3ax-mp 5 . . 3 (1o𝑦) ∈ V
5 fvex 6676 . . 3 (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V
64, 5ifex 4511 . 2 if(𝑥 = ∅, (1o𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)) ∈ V
71, 6fnmpoi 7757 1 o Fn (On × On)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  c0 4288  ifcif 4463  cmpt 5137   × cxp 5546  Oncon0 6184   Fn wfn 6343  cfv 6348  (class class class)co 7145  reccrdg 8034  1oc1o 8084   ·o comu 8089  o coe 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-1o 8091  df-oexp 8097
This theorem is referenced by:  oaabs2  8261  omabs  8263
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