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Theorem cdafn 7811
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 7810 . 2  |-  +c  =  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
2 vex 2804 . . . 4  |-  x  e. 
_V
3 snex 4232 . . . 4  |-  { (/) }  e.  _V
42, 3xpex 4817 . . 3  |-  ( x  X.  { (/) } )  e.  _V
5 vex 2804 . . . 4  |-  y  e. 
_V
6 snex 4232 . . . 4  |-  { 1o }  e.  _V
75, 6xpex 4817 . . 3  |-  ( y  X.  { 1o }
)  e.  _V
84, 7unex 4534 . 2  |-  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o }
) )  e.  _V
91, 8fnmpt2i 6209 1  |-  +c  Fn  ( _V  X.  _V )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2801    u. cun 3163   (/)c0 3468   {csn 3653    X. cxp 4703    Fn wfn 5266   1oc1o 6488    +c ccda 7809
This theorem is referenced by:  cda1dif  7818  cdacomen  7823  cdadom1  7828  cdainf  7834  pwcdadom  7858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-cda 7810
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