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Theorem rankvalg 7732
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7731 expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003.)
Assertion
Ref Expression
rankvalg  |-  ( A  e.  V  ->  ( rank `  A )  = 
|^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem rankvalg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fveq2 5719 . . 3  |-  ( y  =  A  ->  ( rank `  y )  =  ( rank `  A
) )
2 eleq1 2495 . . . . 5  |-  ( y  =  A  ->  (
y  e.  ( R1
`  suc  x )  <->  A  e.  ( R1 `  suc  x ) ) )
32rabbidv 2940 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
43inteqd 4047 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
51, 4eqeq12d 2449 . 2  |-  ( y  =  A  ->  (
( rank `  y )  =  |^| { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  <->  ( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) } ) )
6 vex 2951 . . 3  |-  y  e. 
_V
76rankval 7731 . 2  |-  ( rank `  y )  =  |^| { x  e.  On  | 
y  e.  ( R1
`  suc  x ) }
85, 7vtoclg 3003 1  |-  ( A  e.  V  ->  ( rank `  A )  = 
|^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2701   |^|cint 4042   Oncon0 4573   suc csuc 4575   ` cfv 5445   R1cr1 7677   rankcrnk 7678
This theorem is referenced by:  rankval2  7733
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-reg 7549  ax-inf2 7585
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-recs 6624  df-rdg 6659  df-r1 7679  df-rank 7680
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