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Theorem rankvalb 8293
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 8312 does not use Regularity, and so requires the assumption that  A is in the range of  R1. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
Assertion
Ref Expression
rankvalb  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
Distinct variable group:    x, A

Proof of Theorem rankvalb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 3065 . 2  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  _V )
2 rankwflemb 8289 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
3 intexrab 4575 . . 3  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
42, 3sylbb 202 . 2  |-  ( A  e.  U. ( R1
" On )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
5 eleq1 2527 . . . . 5  |-  ( y  =  A  ->  (
y  e.  ( R1
`  suc  x )  <->  A  e.  ( R1 `  suc  x ) ) )
65rabbidv 3047 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
76inteqd 4252 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
8 df-rank 8261 . . 3  |-  rank  =  ( y  e.  _V  |->  |^|
{ x  e.  On  |  y  e.  ( R1 `  suc  x ) } )
97, 8fvmptg 5968 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )  ->  ( rank `  A
)  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
101, 4, 9syl2anc 671 1  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454    e. wcel 1897   E.wrex 2749   {crab 2752   _Vcvv 3056   U.cuni 4211   |^|cint 4247   "cima 4855   Oncon0 5441   suc csuc 5443   ` cfv 5600   R1cr1 8258   rankcrnk 8259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-r1 8260  df-rank 8261
This theorem is referenced by:  rankr1ai  8294  rankidb  8296  rankval  8312
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