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Theorem r1suc 7437
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 7436 . 2  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
2 r1fnon 7434 . . . 4  |-  R1  Fn  On
3 fndm 5308 . . . 4  |-  ( R1  Fn  On  ->  dom  R1  =  On )
42, 3ax-mp 10 . . 3  |-  dom  R1  =  On
54eqcomi 2288 . 2  |-  On  =  dom  R1
61, 5eleq2s 2376 1  |-  ( A  e.  On  ->  ( R1 `  suc  A )  =  ~P ( R1
`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1628    e. wcel 1688   ~Pcpw 3626   Oncon0 4391   suc csuc 4393   dom cdm 4688    Fn wfn 5216   ` cfv 5221   R1cr1 7429
This theorem is referenced by:  r1sdom  7441  r1sssuc  7450  tz9.12lem3  7456  rankval2  7485  rankpwi  7490  dfac12lem2  7765  dfac12r  7767  ackbij2lem2  7861  ackbij2lem3  7862  wunr1om  8336  r1wunlim  8354  tskr1om  8384  inar1  8392  inatsk  8395  grur1a  8436  grothomex  8446  rankeq1o  24208  elhf2  24212  0hf  24214  aomclem1  26550
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6383  df-rdg 6418  df-r1 7431
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