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Theorem r1suc 7458
Description: Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
Assertion
Ref Expression
r1suc  |-  ( A  e.  On  ->  ( R1 `  suc  A )  =  ~P ( R1
`  A ) )

Proof of Theorem r1suc
StepHypRef Expression
1 r1sucg 7457 . 2  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
2 r1fnon 7455 . . . 4  |-  R1  Fn  On
3 fndm 5359 . . . 4  |-  ( R1  Fn  On  ->  dom  R1  =  On )
42, 3ax-mp 8 . . 3  |-  dom  R1  =  On
54eqcomi 2300 . 2  |-  On  =  dom  R1
61, 5eleq2s 2388 1  |-  ( A  e.  On  ->  ( R1 `  suc  A )  =  ~P ( R1
`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ~Pcpw 3638   Oncon0 4408   suc csuc 4410   dom cdm 4705    Fn wfn 5266   ` cfv 5271   R1cr1 7450
This theorem is referenced by:  r1sdom  7462  r1sssuc  7471  tz9.12lem3  7477  rankval2  7506  rankpwi  7511  dfac12lem2  7786  dfac12r  7788  ackbij2lem2  7882  ackbij2lem3  7883  wunr1om  8357  r1wunlim  8375  tskr1om  8405  inar1  8413  inatsk  8416  grur1a  8457  grothomex  8467  rankeq1o  24873  elhf2  24877  0hf  24879  aomclem1  27254
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-rep 4147  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-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-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
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