MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnoa Structured version   Unicode version

Theorem fnoa 7209
Description: Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fnoa  |-  +o  Fn  ( On  X.  On )

Proof of Theorem fnoa
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-oadd 7185 . 2  |-  +o  =  ( x  e.  On ,  y  e.  On  |->  ( rec ( ( z  e.  _V  |->  suc  z
) ,  x ) `
 y ) )
2 fvex 5882 . 2  |-  ( rec ( ( z  e. 
_V  |->  suc  z ) ,  x ) `  y
)  e.  _V
31, 2fnmpt2i 6867 1  |-  +o  Fn  ( On  X.  On )
Colors of variables: wff setvar class
Syntax hints:   _Vcvv 3078    |-> cmpt 4475    X. cxp 4843   Oncon0 5433   suc csuc 5435    Fn wfn 5587   ` cfv 5592   reccrdg 7126    +o coa 7178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-oadd 7185
This theorem is referenced by:  cantnfvalf  8160  dmaddpi  9304
  Copyright terms: Public domain W3C validator