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Theorem rankidb 7488
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 7481 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
2 nfcv 2432 . . . . . 6  |-  F/_ x R1
3 nfrab1 2733 . . . . . . . 8  |-  F/_ x { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
43nfint 3888 . . . . . . 7  |-  F/_ x |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
54nfsuc 4479 . . . . . 6  |-  F/_ x  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
62, 5nffv 5548 . . . . 5  |-  F/_ x
( R1 `  suc  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
76nfel2 2444 . . . 4  |-  F/ x  A  e.  ( R1 ` 
suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
8 suceq 4473 . . . . . 6  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  suc  x  =  suc  |^| { x  e.  On  |  A  e.  ( R1 ` 
suc  x ) } )
98fveq2d 5545 . . . . 5  |-  ( x  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  ( R1 `  suc  x
)  =  ( R1
`  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
109eleq2d 2363 . . . 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 4606 . . 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 7485 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
14 suceq 4473 . . . 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 5545 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( R1 `  suc  ( rank `  A )
)  =  ( R1
`  suc  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } ) )
1712, 16eleqtrrd 2373 1  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560   U.cuni 3843   |^|cint 3878   Oncon0 4408   suc csuc 4410   "cima 4708   ` cfv 5271   R1cr1 7450   rankcrnk 7451
This theorem is referenced by:  rankdmr1  7489  rankr1ag  7490  sswf  7496  uniwf  7507  rankonidlem  7516  rankid  7521  dfac12lem2  7786  aomclem4  27257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-r1 7452  df-rank 7453
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