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Theorem rankvalb 7353
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7372 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
StepHypRef Expression
1 elex 2735 . 2  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  _V )
2 rankwflemb 7349 . . . 4  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
3 intexrab 4068 . . . 4  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
42, 3bitri 242 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
54biimpi 188 . 2  |-  ( A  e.  U. ( R1
" On )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
6 eleq1 2313 . . . . 5  |-  ( y  =  A  ->  (
y  e.  ( R1
`  suc  x )  <->  A  e.  ( R1 `  suc  x ) ) )
76rabbidv 2719 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
87inteqd 3765 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
9 df-rank 7321 . . 3  |-  rank  =  ( y  e.  _V  |->  |^|
{ x  e.  On  |  y  e.  ( R1 `  suc  x ) } )
108, 9fvmptg 5452 . 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 645 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 6    = wceq 1619    e. wcel 1621   E.wrex 2510   {crab 2512   _Vcvv 2727   U.cuni 3727   |^|cint 3760   Oncon0 4285   suc csuc 4287   "cima 4583   ` cfv 4592   R1cr1 7318   rankcrnk 7319
This theorem is referenced by:  rankr1ai  7354  rankidb  7356  rankval  7372
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-recs 6274  df-rdg 6309  df-r1 7320  df-rank 7321
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