MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankvalg Unicode version

Theorem rankvalg 7443
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7442 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
StepHypRef Expression
1 fveq2 5444 . . 3  |-  ( y  =  A  ->  ( rank `  y )  =  ( rank `  A
) )
2 eleq1 2316 . . . . 5  |-  ( y  =  A  ->  (
y  e.  ( R1
`  suc  x )  <->  A  e.  ( R1 `  suc  x ) ) )
32rabbidv 2749 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
43inteqd 3827 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
51, 4eqeq12d 2270 . 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 2760 . . 3  |-  y  e. 
_V
76rankval 7442 . 2  |-  ( rank `  y )  =  |^| { x  e.  On  | 
y  e.  ( R1
`  suc  x ) }
85, 7vtoclg 2811 1  |-  ( A  e.  V  ->  ( 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   {crab 2520   |^|cint 3822   Oncon0 4350   suc csuc 4352   ` cfv 4659   R1cr1 7388   rankcrnk 7389
This theorem is referenced by:  rankval2  7444
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 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-reg 7260  ax-inf2 7296
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-recs 6342  df-rdg 6377  df-r1 7390  df-rank 7391
  Copyright terms: Public domain W3C validator