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Theorem rankval 7742
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). (Contributed by NM, 24-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
Hypothesis
Ref Expression
rankval.1  |-  A  e. 
_V
Assertion
Ref Expression
rankval  |-  ( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) }
Distinct variable group:    x, A

Proof of Theorem rankval
StepHypRef Expression
1 rankval.1 . . 3  |-  A  e. 
_V
2 unir1 7739 . . 3  |-  U. ( R1 " On )  =  _V
31, 2eleqtrri 2509 . 2  |-  A  e. 
U. ( R1 " On )
4 rankvalb 7723 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
53, 4ax-mp 8 1  |-  ( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) }
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956   U.cuni 4015   |^|cint 4050   Oncon0 4581   suc csuc 4583   "cima 4881   ` cfv 5454   R1cr1 7688   rankcrnk 7689
This theorem is referenced by:  rankvalg  7743  rankeq1o  26112
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-reg 7560  ax-inf2 7596
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-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-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  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-recs 6633  df-rdg 6668  df-r1 7690  df-rank 7691
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