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Theorem rankval 8028
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 8025 . . 3  |-  U. ( R1 " On )  =  _V
31, 2eleqtrri 2516 . 2  |-  A  e. 
U. ( R1 " On )
4 rankvalb 8009 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
53, 4ax-mp 5 1  |-  ( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   {crab 2724   _Vcvv 2977   U.cuni 4096   |^|cint 4133   Oncon0 4724   suc csuc 4726   "cima 4848   ` cfv 5423   R1cr1 7974   rankcrnk 7975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-reg 7812  ax-inf2 7852
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-om 6482  df-recs 6837  df-rdg 6871  df-r1 7976  df-rank 7977
This theorem is referenced by:  rankvalg  8029  rankeq1o  28214
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