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Theorem fnoe 7635
 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 7611 . 2 𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1𝑜𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)))
2 1on 7612 . . . 4 1𝑜 ∈ On
3 difexg 4841 . . . 4 (1𝑜 ∈ On → (1𝑜𝑦) ∈ V)
42, 3ax-mp 5 . . 3 (1𝑜𝑦) ∈ V
5 fvex 6239 . . 3 (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦) ∈ V
64, 5ifex 4189 . 2 if(𝑥 = ∅, (1𝑜𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)) ∈ V
71, 6fnmpt2i 7284 1 𝑜 Fn (On × On)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604  ∅c0 3948  ifcif 4119   ↦ cmpt 4762   × cxp 5141  Oncon0 5761   Fn wfn 5921  ‘cfv 5926  (class class class)co 6690  reccrdg 7550  1𝑜c1o 7598   ·𝑜 comu 7603   ↑𝑜 coe 7604 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-1o 7605  df-oexp 7611 This theorem is referenced by:  oaabs2  7770  omabs  7772
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