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Theorem cardf2 7822
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 7818 . . . 4  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
21funmpt2 5482 . . 3  |-  Fun  card
3 rabab 2965 . . . 4  |-  { x  e.  _V  |  |^| { y  e.  On  |  y 
~~  x }  e.  _V }  =  { x  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
41dmmpt 5357 . . . 4  |-  dom  card  =  { x  e.  _V  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
5 intexrab 4351 . . . . 5  |-  ( E. y  e.  On  y  ~~  x  <->  |^| { y  e.  On  |  y  ~~  x }  e.  _V )
65abbii 2547 . . . 4  |-  { x  |  E. y  e.  On  y  ~~  x }  =  { x  |  |^| { y  e.  On  | 
y  ~~  x }  e.  _V }
73, 4, 63eqtr4i 2465 . . 3  |-  dom  card  =  { x  |  E. y  e.  On  y  ~~  x }
8 df-fn 5449 . . 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 2951 . . . . . . . . 9  |-  w  e. 
_V
1210, 11syl6eqelr 2524 . . . . . . . 8  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  |^| { y  e.  On  |  y  ~~  z }  e.  _V )
13 intex 4348 . . . . . . . 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 3639 . . . . . . 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 2951 . . . . . . 7  |-  z  e. 
_V
18 breq2 4208 . . . . . . . 8  |-  ( x  =  z  ->  (
y  ~~  x  <->  y  ~~  z ) )
1918rexbidv 2718 . . . . . . 7  |-  ( x  =  z  ->  ( E. y  e.  On  y  ~~  x  <->  E. y  e.  On  y  ~~  z
) )
2017, 19elab 3074 . . . . . 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 3420 . . . . . . 7  |-  { y  e.  On  |  y 
~~  z }  C_  On
23 oninton 4772 . . . . . . 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 2509 . . . . 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 4474 . . 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 7818 . . . 4  |-  card  =  ( z  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  z } )
29 df-mpt 4260 . . . 4  |-  ( z  e.  _V  |->  |^| { y  e.  On  |  y 
~~  z } )  =  { <. z ,  w >.  |  (
z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } ) }
3028, 29eqtri 2455 . . 3  |-  card  =  { <. z ,  w >.  |  ( z  e. 
_V  /\  w  =  |^| { y  e.  On  |  y  ~~  z } ) }
31 df-xp 4876 . . 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 3379 . 2  |-  card  C_  ( { x  |  E. y  e.  On  y  ~~  x }  X.  On )
33 dff2 5873 . 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 1652    e. wcel 1725   {cab 2421    =/= wne 2598   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   (/)c0 3620   |^|cint 4042   class class class wbr 4204   {copab 4257    e. cmpt 4258   Oncon0 4573    X. cxp 4868   dom cdm 4870   Fun wfun 5440    Fn wfn 5441   -->wf 5442    ~~ cen 7098   cardccrd 7814
This theorem is referenced by:  cardon  7823  isnum2  7824  cardf  8417  smobeth  8453  hashkf  11612  hashgval  11613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-card 7818
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