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Theorem r1suc 7442
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 7441 . 2  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
2 r1fnon 7439 . . . 4  |-  R1  Fn  On
3 fndm 5343 . . . 4  |-  ( R1  Fn  On  ->  dom  R1  =  On )
42, 3ax-mp 8 . . 3  |-  dom  R1  =  On
54eqcomi 2287 . 2  |-  On  =  dom  R1
61, 5eleq2s 2375 1  |-  ( A  e.  On  ->  ( R1 `  suc  A )  =  ~P ( R1
`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ~Pcpw 3625   Oncon0 4392   suc csuc 4394   dom cdm 4689    Fn wfn 5250   ` cfv 5255   R1cr1 7434
This theorem is referenced by:  r1sdom  7446  r1sssuc  7455  tz9.12lem3  7461  rankval2  7490  rankpwi  7495  dfac12lem2  7770  dfac12r  7772  ackbij2lem2  7866  ackbij2lem3  7867  wunr1om  8341  r1wunlim  8359  tskr1om  8389  inar1  8397  inatsk  8400  grur1a  8441  grothomex  8451  rankeq1o  24801  elhf2  24805  0hf  24807  aomclem1  27151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436
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