MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardf2 Structured version   Unicode version

Theorem cardf2 8227
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 8223 . . . 4  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
21funmpt2 5566 . . 3  |-  Fun  card
3 rabab 3096 . . . 4  |-  { x  e.  _V  |  |^| { y  e.  On  |  y 
~~  x }  e.  _V }  =  { x  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
41dmmpt 5444 . . . 4  |-  dom  card  =  { x  e.  _V  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
5 intexrab 4562 . . . . 5  |-  ( E. y  e.  On  y  ~~  x  <->  |^| { y  e.  On  |  y  ~~  x }  e.  _V )
65abbii 2588 . . . 4  |-  { x  |  E. y  e.  On  y  ~~  x }  =  { x  |  |^| { y  e.  On  | 
y  ~~  x }  e.  _V }
73, 4, 63eqtr4i 2493 . . 3  |-  dom  card  =  { x  |  E. y  e.  On  y  ~~  x }
8 df-fn 5532 . . 3  |-  ( card 
Fn  { x  |  E. y  e.  On  y  ~~  x }  <->  ( Fun  card  /\  dom  card  =  {
x  |  E. y  e.  On  y  ~~  x } ) )
92, 7, 8mpbir2an 911 . 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 3081 . . . . . . . . 9  |-  w  e. 
_V
1210, 11syl6eqelr 2551 . . . . . . . 8  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  |^| { y  e.  On  |  y  ~~  z }  e.  _V )
13 intex 4559 . . . . . . . 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 3768 . . . . . . 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 3081 . . . . . . 7  |-  z  e. 
_V
18 breq2 4407 . . . . . . . 8  |-  ( x  =  z  ->  (
y  ~~  x  <->  y  ~~  z ) )
1918rexbidv 2868 . . . . . . 7  |-  ( x  =  z  ->  ( E. y  e.  On  y  ~~  x  <->  E. y  e.  On  y  ~~  z
) )
2017, 19elab 3213 . . . . . 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 3548 . . . . . . 7  |-  { y  e.  On  |  y 
~~  z }  C_  On
23 oninton 6524 . . . . . . 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 2542 . . . . 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 4727 . . 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 8223 . . . 4  |-  card  =  ( z  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  z } )
29 df-mpt 4463 . . . 4  |-  ( z  e.  _V  |->  |^| { y  e.  On  |  y 
~~  z } )  =  { <. z ,  w >.  |  (
z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } ) }
3028, 29eqtri 2483 . . 3  |-  card  =  { <. z ,  w >.  |  ( z  e. 
_V  /\  w  =  |^| { y  e.  On  |  y  ~~  z } ) }
31 df-xp 4957 . . 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 3506 . 2  |-  card  C_  ( { x  |  E. y  e.  On  y  ~~  x }  X.  On )
33 dff2 5967 . 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 911 1  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439    =/= wne 2648   E.wrex 2800   {crab 2803   _Vcvv 3078    C_ wss 3439   (/)c0 3748   |^|cint 4239   class class class wbr 4403   {copab 4460    |-> cmpt 4461   Oncon0 4830    X. cxp 4949   dom cdm 4951   Fun wfun 5523    Fn wfn 5524   -->wf 5525    ~~ cen 7420   cardccrd 8219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-fun 5531  df-fn 5532  df-f 5533  df-card 8223
This theorem is referenced by:  cardon  8228  isnum2  8229  cardf  8828  smobeth  8864  hashkf  12225  hashgval  12226
  Copyright terms: Public domain W3C validator