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Theorem cardf 8926
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 8325 . 2  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
21fdmi 5736 . . . 4  |-  dom  card  =  { x  |  E. y  e.  On  y  ~~  x }
3 cardeqv 8850 . . . 4  |-  dom  card  =  _V
42, 3eqtr3i 2498 . . 3  |-  { x  |  E. y  e.  On  y  ~~  x }  =  _V
54feq2i 5724 . 2  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  <->  card : _V --> On )
61, 5mpbi 208 1  |-  card : _V --> On
Colors of variables: wff setvar class
Syntax hints:   {cab 2452   E.wrex 2815   _Vcvv 3113   class class class wbr 4447   Oncon0 4878   dom cdm 4999   -->wf 5584    ~~ cen 7514   cardccrd 8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-ac2 8844
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-recs 7043  df-en 7518  df-card 8321  df-ac 8498
This theorem is referenced by:  inar1  9154
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