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

Theorem cardf2 7592
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 7588 . . . 4  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
21funmpt2 5307 . . 3  |-  Fun  card
3 rabab 2818 . . . 4  |-  { x  e.  _V  |  |^| { y  e.  On  |  y 
~~  x }  e.  _V }  =  { x  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
41dmmpt 5184 . . . 4  |-  dom  card  =  { x  e.  _V  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
5 intexrab 4186 . . . . 5  |-  ( E. y  e.  On  y  ~~  x  <->  |^| { y  e.  On  |  y  ~~  x }  e.  _V )
65abbii 2408 . . . 4  |-  { x  |  E. y  e.  On  y  ~~  x }  =  { x  |  |^| { y  e.  On  | 
y  ~~  x }  e.  _V }
73, 4, 63eqtr4i 2326 . . 3  |-  dom  card  =  { x  |  E. y  e.  On  y  ~~  x }
8 df-fn 5274 . . 3  |-  ( card 
Fn  { x  |  E. y  e.  On  y  ~~  x }  <->  ( Fun  card  /\  dom  card  =  {
x  |  E. y  e.  On  y  ~~  x } ) )
92, 7, 8mpbir2an 886 . 2  |-  card  Fn  { x  |  E. y  e.  On  y  ~~  x }
10 simpr 447 . . . . . . . . 9  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  w  =  |^| { y  e.  On  | 
y  ~~  z }
)
11 vex 2804 . . . . . . . . 9  |-  w  e. 
_V
1210, 11syl6eqelr 2385 . . . . . . . 8  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  |^| { y  e.  On  |  y  ~~  z }  e.  _V )
13 intex 4183 . . . . . . . 8  |-  ( { y  e.  On  | 
y  ~~  z }  =/=  (/)  <->  |^| { y  e.  On  |  y  ~~  z }  e.  _V )
1412, 13sylibr 203 . . . . . . 7  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  { y  e.  On  |  y  ~~  z }  =/=  (/) )
15 rabn0 3487 . . . . . . 7  |-  ( { y  e.  On  | 
y  ~~  z }  =/=  (/)  <->  E. y  e.  On  y  ~~  z )
1614, 15sylib 188 . . . . . 6  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  E. y  e.  On  y  ~~  z )
17 vex 2804 . . . . . . 7  |-  z  e. 
_V
18 breq2 4043 . . . . . . . 8  |-  ( x  =  z  ->  (
y  ~~  x  <->  y  ~~  z ) )
1918rexbidv 2577 . . . . . . 7  |-  ( x  =  z  ->  ( E. y  e.  On  y  ~~  x  <->  E. y  e.  On  y  ~~  z
) )
2017, 19elab 2927 . . . . . 6  |-  ( z  e.  { x  |  E. y  e.  On  y  ~~  x }  <->  E. y  e.  On  y  ~~  z
)
2116, 20sylibr 203 . . . . 5  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  z  e.  {
x  |  E. y  e.  On  y  ~~  x } )
22 ssrab2 3271 . . . . . . 7  |-  { y  e.  On  |  y 
~~  z }  C_  On
23 oninton 4607 . . . . . . 7  |-  ( ( { y  e.  On  |  y  ~~  z } 
C_  On  /\  { y  e.  On  |  y 
~~  z }  =/=  (/) )  ->  |^| { y  e.  On  |  y 
~~  z }  e.  On )
2422, 14, 23sylancr 644 . . . . . 6  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  |^| { y  e.  On  |  y  ~~  z }  e.  On )
2510, 24eqeltrd 2370 . . . . 5  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  w  e.  On )
2621, 25jca 518 . . . 4  |-  ( ( z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } )  ->  ( z  e. 
{ x  |  E. y  e.  On  y  ~~  x }  /\  w  e.  On ) )
2726ssopab2i 4308 . . 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 7588 . . . 4  |-  card  =  ( z  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  z } )
29 df-mpt 4095 . . . 4  |-  ( z  e.  _V  |->  |^| { y  e.  On  |  y 
~~  z } )  =  { <. z ,  w >.  |  (
z  e.  _V  /\  w  =  |^| { y  e.  On  |  y 
~~  z } ) }
3028, 29eqtri 2316 . . 3  |-  card  =  { <. z ,  w >.  |  ( z  e. 
_V  /\  w  =  |^| { y  e.  On  |  y  ~~  z } ) }
31 df-xp 4711 . . 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 3230 . 2  |-  card  C_  ( { x  |  E. y  e.  On  y  ~~  x }  X.  On )
33 dff2 5688 . 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 886 1  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   |^|cint 3878   class class class wbr 4039   {copab 4092    e. cmpt 4093   Oncon0 4408    X. cxp 4703   dom cdm 4705   Fun wfun 5265    Fn wfn 5266   -->wf 5267    ~~ cen 6876   cardccrd 7584
This theorem is referenced by:  cardon  7593  isnum2  7594  cardf  8188  smobeth  8224  hashkf  11355  hashgval  11356
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-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  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-fun 5273  df-fn 5274  df-f 5275  df-card 7588
  Copyright terms: Public domain W3C validator