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Theorem rankval 7490
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 7487 . . 3  |-  U. ( R1 " On )  =  _V
31, 2eleqtrri 2358 . 2  |-  A  e. 
U. ( R1 " On )
4 rankvalb 7471 . 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 1625    e. wcel 1686   {crab 2549   _Vcvv 2790   U.cuni 3829   |^|cint 3864   Oncon0 4394   suc csuc 4396   "cima 4694   ` cfv 5257   R1cr1 7436   rankcrnk 7437
This theorem is referenced by:  rankvalg  7491  rankeq1o  24803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-reg 7308  ax-inf2 7344
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425  df-r1 7438  df-rank 7439
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