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Theorem cdafn 8617
Description: Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
cdafn  |-  +c  Fn  ( _V  X.  _V )

Proof of Theorem cdafn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cda 8616 . 2  |-  +c  =  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
2 vex 3034 . . . 4  |-  x  e. 
_V
3 snex 4641 . . . 4  |-  { (/) }  e.  _V
42, 3xpex 6614 . . 3  |-  ( x  X.  { (/) } )  e.  _V
5 vex 3034 . . . 4  |-  y  e. 
_V
6 snex 4641 . . . 4  |-  { 1o }  e.  _V
75, 6xpex 6614 . . 3  |-  ( y  X.  { 1o }
)  e.  _V
84, 7unex 6608 . 2  |-  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o }
) )  e.  _V
91, 8fnmpt2i 6881 1  |-  +c  Fn  ( _V  X.  _V )
Colors of variables: wff setvar class
Syntax hints:   _Vcvv 3031    u. cun 3388   (/)c0 3722   {csn 3959    X. cxp 4837    Fn wfn 5584   1oc1o 7193    +c ccda 8615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-cda 8616
This theorem is referenced by:  cda1dif  8624  cdacomen  8629  cdadom1  8634  cdainf  8640  pwcdadom  8664
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