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Theorem rankidb 7472
Description: Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
Assertion
Ref Expression
rankidb  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )

Proof of Theorem rankidb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rankwflemb 7465 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
2 nfcv 2419 . . . . . 6  |-  F/_ x R1
3 nfrab1 2720 . . . . . . . 8  |-  F/_ x { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
43nfint 3872 . . . . . . 7  |-  F/_ x |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
54nfsuc 4463 . . . . . 6  |-  F/_ x  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
62, 5nffv 5532 . . . . 5  |-  F/_ x
( R1 `  suc  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
76nfel2 2431 . . . 4  |-  F/ x  A  e.  ( R1 ` 
suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
8 suceq 4457 . . . . . 6  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  suc  x  =  suc  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) } )
98fveq2d 5529 . . . . 5  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  ( R1 `  suc  x
)  =  ( R1
`  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
109eleq2d 2350 . . . 4  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  ( A  e.  ( R1
`  suc  x )  <->  A  e.  ( R1 `  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) ) )
117, 10onminsb 4590 . . 3  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  ->  A  e.  ( R1 `  suc  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
121, 11sylbi 187 . 2  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
13 rankvalb 7469 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
14 suceq 4457 . . . 4  |-  ( (
rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  suc  ( rank `  A
)  =  suc  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) } )
1513, 14syl 15 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  suc  ( rank `  A
)  =  suc  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) } )
1615fveq2d 5529 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( R1 `  suc  ( rank `  A )
)  =  ( R1
`  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
1712, 16eleqtrrd 2360 1  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   U.cuni 3827   |^|cint 3862   Oncon0 4392   suc csuc 4394   "cima 4692   ` cfv 5255   R1cr1 7434   rankcrnk 7435
This theorem is referenced by:  rankdmr1  7473  rankr1ag  7474  sswf  7480  uniwf  7491  rankonidlem  7500  rankid  7505  dfac12lem2  7770  aomclem4  27154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
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