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

Theorem fnoe 7535
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 7511 . 2 𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1𝑜𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)))
2 1on 7512 . . . 4 1𝑜 ∈ On
3 difexg 4768 . . . 4 (1𝑜 ∈ On → (1𝑜𝑦) ∈ V)
42, 3ax-mp 5 . . 3 (1𝑜𝑦) ∈ V
5 fvex 6158 . . 3 (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦) ∈ V
64, 5ifex 4128 . 2 if(𝑥 = ∅, (1𝑜𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)) ∈ V
71, 6fnmpt2i 7184 1 𝑜 Fn (On × On)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  c0 3891  ifcif 4058  cmpt 4673   × cxp 5072  Oncon0 5682   Fn wfn 5842  cfv 5847  (class class class)co 6604  reccrdg 7450  1𝑜c1o 7498   ·𝑜 comu 7503  𝑜 coe 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-1o 7505  df-oexp 7511
This theorem is referenced by:  oaabs2  7670  omabs  7672
  Copyright terms: Public domain W3C validator