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Theorem rankidb 7690
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 7683 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
2 nfcv 2548 . . . . . 6  |-  F/_ x R1
3 nfrab1 2856 . . . . . . . 8  |-  F/_ x { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
43nfint 4028 . . . . . . 7  |-  F/_ x |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
54nfsuc 4620 . . . . . 6  |-  F/_ x  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
62, 5nffv 5702 . . . . 5  |-  F/_ x
( R1 `  suc  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
76nfel2 2560 . . . 4  |-  F/ x  A  e.  ( R1 ` 
suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
8 suceq 4614 . . . . . 6  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  suc  x  =  suc  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) } )
98fveq2d 5699 . . . . 5  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  ( R1 `  suc  x
)  =  ( R1
`  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
109eleq2d 2479 . . . 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 4746 . . 3  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  ->  A  e.  ( R1 `  suc  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
121, 11sylbi 188 . 2  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
13 rankvalb 7687 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
14 suceq 4614 . . . 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 5699 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( R1 `  suc  ( rank `  A )
)  =  ( R1
`  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
1712, 16eleqtrrd 2489 1  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   E.wrex 2675   {crab 2678   U.cuni 3983   |^|cint 4018   Oncon0 4549   suc csuc 4551   "cima 4848   ` cfv 5421   R1cr1 7652   rankcrnk 7653
This theorem is referenced by:  rankdmr1  7691  rankr1ag  7692  sswf  7698  uniwf  7709  rankonidlem  7718  rankid  7723  dfac12lem2  7988  aomclem4  27030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-recs 6600  df-rdg 6635  df-r1 7654  df-rank 7655
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