Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnoe Structured version   Visualization version   GIF version

Theorem fnoe 8113
 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 8086 . 2 o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
2 1on 8087 . . . 4 1o ∈ On
3 difexg 5207 . . . 4 (1o ∈ On → (1o𝑦) ∈ V)
42, 3ax-mp 5 . . 3 (1o𝑦) ∈ V
5 fvex 6659 . . 3 (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V
64, 5ifex 4491 . 2 if(𝑥 = ∅, (1o𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)) ∈ V
71, 6fnmpoi 7746 1 o Fn (On × On)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1537   ∈ wcel 2114  Vcvv 3473   ∖ cdif 3910  ∅c0 4269  ifcif 4443   ↦ cmpt 5122   × cxp 5529  Oncon0 6167   Fn wfn 6326  ‘cfv 6331  (class class class)co 7133  reccrdg 8023  1oc1o 8073   ·o comu 8078   ↑o coe 8079 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-ord 6170  df-on 6171  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fv 6339  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-1o 8080  df-oexp 8086 This theorem is referenced by:  oaabs2  8250  omabs  8252
 Copyright terms: Public domain W3C validator