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Theorem r1suc 7685
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 7684 . 2  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
2 r1fnon 7682 . . . 4  |-  R1  Fn  On
3 fndm 5535 . . . 4  |-  ( R1  Fn  On  ->  dom  R1  =  On )
42, 3ax-mp 8 . . 3  |-  dom  R1  =  On
54eqcomi 2439 . 2  |-  On  =  dom  R1
61, 5eleq2s 2527 1  |-  ( A  e.  On  ->  ( R1 `  suc  A )  =  ~P ( R1
`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ~Pcpw 3791   Oncon0 4573   suc csuc 4575   dom cdm 4869    Fn wfn 5440   ` cfv 5445   R1cr1 7677
This theorem is referenced by:  r1sdom  7689  r1sssuc  7698  tz9.12lem3  7704  rankval2  7733  rankpwi  7738  dfac12lem2  8013  dfac12r  8015  ackbij2lem2  8109  ackbij2lem3  8110  wunr1om  8583  r1wunlim  8601  tskr1om  8631  inar1  8639  inatsk  8642  grur1a  8683  grothomex  8693  rankeq1o  26060  elhf2  26064  0hf  26066  aomclem1  27066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-recs 6624  df-rdg 6659  df-r1 7679
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