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Theorem cardf2 8341
Description: The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
cardf2  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
Distinct variable group:    x, y

Proof of Theorem cardf2
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-card 8337 . . . 4  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
21funmpt2 5631 . . 3  |-  Fun  card
3 rabab 3127 . . . 4  |-  { x  e.  _V  |  |^| { y  e.  On  |  y 
~~  x }  e.  _V }  =  { x  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
41dmmpt 5508 . . . 4  |-  dom  card  =  { x  e.  _V  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
5 intexrab 4615 . . . . 5  |-  ( E. y  e.  On  y  ~~  x  <->  |^| { y  e.  On  |  y  ~~  x }  e.  _V )
65abbii 2591 . . . 4  |-  { x  |  E. y  e.  On  y  ~~  x }  =  { x  |  |^| { y  e.  On  | 
y  ~~  x }  e.  _V }
73, 4, 63eqtr4i 2496 . . 3  |-  dom  card  =  { x  |  E. y  e.  On  y  ~~  x }
8 df-fn 5597 . . 3  |-  ( card 
Fn  { x  |  E. y  e.  On  y  ~~  x }  <->  ( Fun  card  /\  dom  card  =  {
x  |  E. y  e.  On  y  ~~  x } ) )
92, 7, 8mpbir2an 920 . 2  |-  card  Fn  { x  |  E. y  e.  On  y  ~~  x }
10 simpr 461 . . . . . . . . 9  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  w  =  |^| { y  e.  On  | 
y  ~~  z }
)
11 vex 3112 . . . . . . . . 9  |-  w  e. 
_V
1210, 11syl6eqelr 2554 . . . . . . . 8  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  |^| { y  e.  On  |  y  ~~  z }  e.  _V )
13 intex 4612 . . . . . . . 8  |-  ( { y  e.  On  | 
y  ~~  z }  =/=  (/)  <->  |^| { y  e.  On  |  y  ~~  z }  e.  _V )
1412, 13sylibr 212 . . . . . . 7  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  { y  e.  On  |  y  ~~  z }  =/=  (/) )
15 rabn0 3814 . . . . . . 7  |-  ( { y  e.  On  | 
y  ~~  z }  =/=  (/)  <->  E. y  e.  On  y  ~~  z )
1614, 15sylib 196 . . . . . 6  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  E. y  e.  On  y  ~~  z )
17 vex 3112 . . . . . . 7  |-  z  e. 
_V
18 breq2 4460 . . . . . . . 8  |-  ( x  =  z  ->  (
y  ~~  x  <->  y  ~~  z ) )
1918rexbidv 2968 . . . . . . 7  |-  ( x  =  z  ->  ( E. y  e.  On  y  ~~  x  <->  E. y  e.  On  y  ~~  z
) )
2017, 19elab 3246 . . . . . 6  |-  ( z  e.  { x  |  E. y  e.  On  y  ~~  x }  <->  E. y  e.  On  y  ~~  z
)
2116, 20sylibr 212 . . . . 5  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  z  e.  {
x  |  E. y  e.  On  y  ~~  x } )
22 ssrab2 3581 . . . . . . 7  |-  { y  e.  On  |  y 
~~  z }  C_  On
23 oninton 6634 . . . . . . 7  |-  ( ( { y  e.  On  |  y  ~~  z } 
C_  On  /\  { y  e.  On  |  y 
~~  z }  =/=  (/) )  ->  |^| { y  e.  On  |  y 
~~  z }  e.  On )
2422, 14, 23sylancr 663 . . . . . 6  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  |^| { y  e.  On  |  y  ~~  z }  e.  On )
2510, 24eqeltrd 2545 . . . . 5  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  w  e.  On )
2621, 25jca 532 . . . 4  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  ( z  e. 
{ x  |  E. y  e.  On  y  ~~  x }  /\  w  e.  On ) )
2726ssopab2i 4784 . . 3  |-  { <. z ,  w >.  |  ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } ) }  C_  { <. z ,  w >.  |  (
z  e.  { x  |  E. y  e.  On  y  ~~  x }  /\  w  e.  On ) }
28 df-card 8337 . . . 4  |-  card  =  ( z  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  z } )
29 df-mpt 4517 . . . 4  |-  ( z  e.  _V  |->  |^| { y  e.  On  |  y 
~~  z } )  =  { <. z ,  w >.  |  (
z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } ) }
3028, 29eqtri 2486 . . 3  |-  card  =  { <. z ,  w >.  |  ( z  e. 
_V  /\  w  =  |^| { y  e.  On  |  y  ~~  z } ) }
31 df-xp 5014 . . 3  |-  ( { x  |  E. y  e.  On  y  ~~  x }  X.  On )  =  { <. z ,  w >.  |  ( z  e. 
{ x  |  E. y  e.  On  y  ~~  x }  /\  w  e.  On ) }
3227, 30, 313sstr4i 3538 . 2  |-  card  C_  ( { x  |  E. y  e.  On  y  ~~  x }  X.  On )
33 dff2 6044 . 2  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  <->  ( card  Fn 
{ x  |  E. y  e.  On  y  ~~  x }  /\  card  C_  ( { x  |  E. y  e.  On  y  ~~  x }  X.  On ) ) )
349, 32, 33mpbir2an 920 1  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442    =/= wne 2652   E.wrex 2808   {crab 2811   _Vcvv 3109    C_ wss 3471   (/)c0 3793   |^|cint 4288   class class class wbr 4456   {copab 4514    |-> cmpt 4515   Oncon0 4887    X. cxp 5006   dom cdm 5008   Fun wfun 5588    Fn wfn 5589   -->wf 5590    ~~ cen 7532   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-fun 5596  df-fn 5597  df-f 5598  df-card 8337
This theorem is referenced by:  cardon  8342  isnum2  8343  cardf  8942  smobeth  8978  hashkf  12410  hashgval  12411
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