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Theorem cardf2 7764
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 7760 . . . 4  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
21funmpt2 5431 . . 3  |-  Fun  card
3 rabab 2917 . . . 4  |-  { x  e.  _V  |  |^| { y  e.  On  |  y 
~~  x }  e.  _V }  =  { x  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
41dmmpt 5306 . . . 4  |-  dom  card  =  { x  e.  _V  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
5 intexrab 4301 . . . . 5  |-  ( E. y  e.  On  y  ~~  x  <->  |^| { y  e.  On  |  y  ~~  x }  e.  _V )
65abbii 2500 . . . 4  |-  { x  |  E. y  e.  On  y  ~~  x }  =  { x  |  |^| { y  e.  On  | 
y  ~~  x }  e.  _V }
73, 4, 63eqtr4i 2418 . . 3  |-  dom  card  =  { x  |  E. y  e.  On  y  ~~  x }
8 df-fn 5398 . . 3  |-  ( card 
Fn  { x  |  E. y  e.  On  y  ~~  x }  <->  ( Fun  card  /\  dom  card  =  {
x  |  E. y  e.  On  y  ~~  x } ) )
92, 7, 8mpbir2an 887 . 2  |-  card  Fn  { x  |  E. y  e.  On  y  ~~  x }
10 simpr 448 . . . . . . . . 9  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  w  =  |^| { y  e.  On  | 
y  ~~  z }
)
11 vex 2903 . . . . . . . . 9  |-  w  e. 
_V
1210, 11syl6eqelr 2477 . . . . . . . 8  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  |^| { y  e.  On  |  y  ~~  z }  e.  _V )
13 intex 4298 . . . . . . . 8  |-  ( { y  e.  On  | 
y  ~~  z }  =/=  (/)  <->  |^| { y  e.  On  |  y  ~~  z }  e.  _V )
1412, 13sylibr 204 . . . . . . 7  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  { y  e.  On  |  y  ~~  z }  =/=  (/) )
15 rabn0 3591 . . . . . . 7  |-  ( { y  e.  On  | 
y  ~~  z }  =/=  (/)  <->  E. y  e.  On  y  ~~  z )
1614, 15sylib 189 . . . . . 6  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  E. y  e.  On  y  ~~  z )
17 vex 2903 . . . . . . 7  |-  z  e. 
_V
18 breq2 4158 . . . . . . . 8  |-  ( x  =  z  ->  (
y  ~~  x  <->  y  ~~  z ) )
1918rexbidv 2671 . . . . . . 7  |-  ( x  =  z  ->  ( E. y  e.  On  y  ~~  x  <->  E. y  e.  On  y  ~~  z
) )
2017, 19elab 3026 . . . . . 6  |-  ( z  e.  { x  |  E. y  e.  On  y  ~~  x }  <->  E. y  e.  On  y  ~~  z
)
2116, 20sylibr 204 . . . . 5  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  z  e.  {
x  |  E. y  e.  On  y  ~~  x } )
22 ssrab2 3372 . . . . . . 7  |-  { y  e.  On  |  y 
~~  z }  C_  On
23 oninton 4721 . . . . . . 7  |-  ( ( { y  e.  On  |  y  ~~  z } 
C_  On  /\  { y  e.  On  |  y 
~~  z }  =/=  (/) )  ->  |^| { y  e.  On  |  y 
~~  z }  e.  On )
2422, 14, 23sylancr 645 . . . . . 6  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  |^| { y  e.  On  |  y  ~~  z }  e.  On )
2510, 24eqeltrd 2462 . . . . 5  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  w  e.  On )
2621, 25jca 519 . . . 4  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  ( z  e. 
{ x  |  E. y  e.  On  y  ~~  x }  /\  w  e.  On ) )
2726ssopab2i 4424 . . 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 7760 . . . 4  |-  card  =  ( z  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  z } )
29 df-mpt 4210 . . . 4  |-  ( z  e.  _V  |->  |^| { y  e.  On  |  y 
~~  z } )  =  { <. z ,  w >.  |  (
z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } ) }
3028, 29eqtri 2408 . . 3  |-  card  =  { <. z ,  w >.  |  ( z  e. 
_V  /\  w  =  |^| { y  e.  On  |  y  ~~  z } ) }
31 df-xp 4825 . . 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 3331 . 2  |-  card  C_  ( { x  |  E. y  e.  On  y  ~~  x }  X.  On )
33 dff2 5821 . 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 887 1  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2374    =/= wne 2551   E.wrex 2651   {crab 2654   _Vcvv 2900    C_ wss 3264   (/)c0 3572   |^|cint 3993   class class class wbr 4154   {copab 4207    e. cmpt 4208   Oncon0 4523    X. cxp 4817   dom cdm 4819   Fun wfun 5389    Fn wfn 5390   -->wf 5391    ~~ cen 7043   cardccrd 7756
This theorem is referenced by:  cardon  7765  isnum2  7766  cardf  8359  smobeth  8395  hashkf  11548  hashgval  11549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-fun 5397  df-fn 5398  df-f 5399  df-card 7760
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