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Theorem fnom 6755
Description: Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fnom  |-  .o  Fn  ( On  X.  On )

Proof of Theorem fnom
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-omul 6731 . 2  |-  .o  =  ( x  e.  On ,  y  e.  On  |->  ( rec ( ( z  e.  _V  |->  ( z  +o  x ) ) ,  (/) ) `  y
) )
2 fvex 5744 . 2  |-  ( rec ( ( z  e. 
_V  |->  ( z  +o  x ) ) ,  (/) ) `  y )  e.  _V
31, 2fnmpt2i 6422 1  |-  .o  Fn  ( On  X.  On )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2958   (/)c0 3630    e. cmpt 4268   Oncon0 4583    X. cxp 4878    Fn wfn 5451   ` cfv 5456  (class class class)co 6083   reccrdg 6669    +o coa 6723    .o comu 6724
This theorem is referenced by:  om0x  6765  dmmulpi  8770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-omul 6731
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