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Theorem rankvalb 7437
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7456 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 2771 . 2  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  _V )
2 rankwflemb 7433 . . . 4  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
3 intexrab 4146 . . . 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 2318 . . . . 5  |-  ( y  =  A  ->  (
y  e.  ( R1
`  suc  x )  <->  A  e.  ( R1 `  suc  x ) ) )
76rabbidv 2755 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
87inteqd 3841 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
9 df-rank 7405 . . 3  |-  rank  =  ( y  e.  _V  |->  |^|
{ x  e.  On  |  y  e.  ( R1 `  suc  x ) } )
108, 9fvmptg 5534 . 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 2519   {crab 2522   _Vcvv 2763   U.cuni 3801   |^|cint 3836   Oncon0 4364   suc csuc 4366   "cima 4664   ` cfv 4673   R1cr1 7402   rankcrnk 7403
This theorem is referenced by:  rankr1ai  7438  rankidb  7440  rankval  7456
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356  df-rdg 6391  df-r1 7404  df-rank 7405
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