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Theorem rankvalb 7683
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7702 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 2928 . 2  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  _V )
2 rankwflemb 7679 . . . 4  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
3 intexrab 4323 . . . 4  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
42, 3bitri 241 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
54biimpi 187 . 2  |-  ( A  e.  U. ( R1
" On )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
6 eleq1 2468 . . . . 5  |-  ( y  =  A  ->  (
y  e.  ( R1
`  suc  x )  <->  A  e.  ( R1 `  suc  x ) ) )
76rabbidv 2912 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
87inteqd 4019 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
9 df-rank 7651 . . 3  |-  rank  =  ( y  e.  _V  |->  |^|
{ x  e.  On  |  y  e.  ( R1 `  suc  x ) } )
108, 9fvmptg 5767 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )  ->  ( rank `  A
)  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
111, 5, 10syl2anc 643 1  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   E.wrex 2671   {crab 2674   _Vcvv 2920   U.cuni 3979   |^|cint 4014   Oncon0 4545   suc csuc 4547   "cima 4844   ` cfv 5417   R1cr1 7648   rankcrnk 7649
This theorem is referenced by:  rankr1ai  7684  rankidb  7686  rankval  7702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-recs 6596  df-rdg 6631  df-r1 7650  df-rank 7651
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