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Theorem fnom 6508
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 6484 . 2  |-  .o  =  ( x  e.  On ,  y  e.  On  |->  ( rec ( ( z  e.  _V  |->  ( z  +o  x ) ) ,  (/) ) `  y
) )
2 fvex 5539 . 2  |-  ( rec ( ( z  e. 
_V  |->  ( z  +o  x ) ) ,  (/) ) `  y )  e.  _V
31, 2fnmpt2i 6193 1  |-  .o  Fn  ( On  X.  On )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2788   (/)c0 3455    e. cmpt 4077   Oncon0 4392    X. cxp 4687    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   reccrdg 6422    +o coa 6476    .o comu 6477
This theorem is referenced by:  om0x  6518  dmmulpi  8515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-omul 6484
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