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Theorem rankidb 8209
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 8202 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
2 nfcv 2616 . . . . . 6  |-  F/_ x R1
3 nfrab1 3035 . . . . . . . 8  |-  F/_ x { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
43nfint 4281 . . . . . . 7  |-  F/_ x |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
54nfsuc 4938 . . . . . 6  |-  F/_ x  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
62, 5nffv 5855 . . . . 5  |-  F/_ x
( R1 `  suc  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
76nfel2 2634 . . . 4  |-  F/ x  A  e.  ( R1 ` 
suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
8 suceq 4932 . . . . . 6  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  suc  x  =  suc  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) } )
98fveq2d 5852 . . . . 5  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  ( R1 `  suc  x
)  =  ( R1
`  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
109eleq2d 2524 . . . 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 6607 . . 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 8206 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
14 suceq 4932 . . . 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 5852 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( R1 `  suc  ( rank `  A )
)  =  ( R1
`  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
1712, 16eleqtrrd 2545 1  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   E.wrex 2805   {crab 2808   U.cuni 4235   |^|cint 4271   Oncon0 4867   suc csuc 4869   "cima 4991   ` cfv 5570   R1cr1 8171   rankcrnk 8172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173  df-rank 8174
This theorem is referenced by:  rankdmr1  8210  rankr1ag  8211  sswf  8217  uniwf  8228  rankonidlem  8237  rankid  8242  dfac12lem2  8515  aomclem4  31242
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