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Theorem cardf 8425
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, 13-Sep-2013.)
Assertion
Ref Expression
cardf  |-  card : _V --> On

Proof of Theorem cardf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardf2 7830 . 2  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
21fdmi 5596 . . . 4  |-  dom  card  =  { x  |  E. y  e.  On  y  ~~  x }
3 cardeqv 8349 . . . 4  |-  dom  card  =  _V
42, 3eqtr3i 2458 . . 3  |-  { x  |  E. y  e.  On  y  ~~  x }  =  _V
54feq2i 5586 . 2  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  <->  card : _V --> On )
61, 5mpbi 200 1  |-  card : _V --> On
Colors of variables: wff set class
Syntax hints:   {cab 2422   E.wrex 2706   _Vcvv 2956   class class class wbr 4212   Oncon0 4581   dom cdm 4878   -->wf 5450    ~~ cen 7106   cardccrd 7822
This theorem is referenced by:  inar1  8650
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-ac2 8343
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-en 7110  df-card 7826  df-ac 7997
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