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Theorem rankidb 8214
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 8207 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
2 nfcv 2629 . . . . . 6  |-  F/_ x R1
3 nfrab1 3042 . . . . . . . 8  |-  F/_ x { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
43nfint 4292 . . . . . . 7  |-  F/_ x |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
54nfsuc 4949 . . . . . 6  |-  F/_ x  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
62, 5nffv 5871 . . . . 5  |-  F/_ x
( R1 `  suc  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
76nfel2 2647 . . . 4  |-  F/ x  A  e.  ( R1 ` 
suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
8 suceq 4943 . . . . . 6  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  suc  x  =  suc  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) } )
98fveq2d 5868 . . . . 5  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  ( R1 `  suc  x
)  =  ( R1
`  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
109eleq2d 2537 . . . 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 6612 . . 3  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  ->  A  e.  ( R1 `  suc  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
121, 11sylbi 195 . 2  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
13 rankvalb 8211 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
14 suceq 4943 . . . 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 16 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  suc  ( rank `  A
)  =  suc  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) } )
1615fveq2d 5868 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( R1 `  suc  ( rank `  A )
)  =  ( R1
`  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
1712, 16eleqtrrd 2558 1  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   U.cuni 4245   |^|cint 4282   Oncon0 4878   suc csuc 4880   "cima 5002   ` cfv 5586   R1cr1 8176   rankcrnk 8177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-recs 7039  df-rdg 7073  df-r1 8178  df-rank 8179
This theorem is referenced by:  rankdmr1  8215  rankr1ag  8216  sswf  8222  uniwf  8233  rankonidlem  8242  rankid  8247  dfac12lem2  8520  aomclem4  30607
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