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Theorem r1sucg 7696
Description: Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1sucg  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )

Proof of Theorem r1sucg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rdgsucg 6682 . . 3  |-  ( A  e.  dom  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )  ->  ( rec (
( x  e.  _V  |->  ~P x ) ,  (/) ) `  suc  A )  =  ( ( x  e.  _V  |->  ~P x
) `  ( rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) ) `  A ) ) )
2 df-r1 7691 . . . 4  |-  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
32dmeqi 5072 . . 3  |-  dom  R1  =  dom  rec ( ( x  e.  _V  |->  ~P x ) ,  (/) )
41, 3eleq2s 2529 . 2  |-  ( A  e.  dom  R1  ->  ( rec ( ( x  e.  _V  |->  ~P x
) ,  (/) ) `  suc  A )  =  ( ( x  e.  _V  |->  ~P x ) `  ( rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) ) `  A ) ) )
52fveq1i 5730 . 2  |-  ( R1
`  suc  A )  =  ( rec (
( x  e.  _V  |->  ~P x ) ,  (/) ) `  suc  A )
6 fvex 5743 . . . 4  |-  ( R1
`  A )  e. 
_V
7 pweq 3803 . . . . 5  |-  ( x  =  ( R1 `  A )  ->  ~P x  =  ~P ( R1 `  A ) )
8 eqid 2437 . . . . 5  |-  ( x  e.  _V  |->  ~P x
)  =  ( x  e.  _V  |->  ~P x
)
96pwex 4383 . . . . 5  |-  ~P ( R1 `  A )  e. 
_V
107, 8, 9fvmpt 5807 . . . 4  |-  ( ( R1 `  A )  e.  _V  ->  (
( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ~P ( R1
`  A ) )
116, 10ax-mp 8 . . 3  |-  ( ( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ~P ( R1
`  A )
122fveq1i 5730 . . . 4  |-  ( R1
`  A )  =  ( rec ( ( x  e.  _V  |->  ~P x ) ,  (/) ) `  A )
1312fveq2i 5732 . . 3  |-  ( ( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ( ( x  e.  _V  |->  ~P x
) `  ( rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) ) `  A ) )
1411, 13eqtr3i 2459 . 2  |-  ~P ( R1 `  A )  =  ( ( x  e. 
_V  |->  ~P x ) `  ( rec ( ( x  e.  _V  |->  ~P x
) ,  (/) ) `  A ) )
154, 5, 143eqtr4g 2494 1  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2957   (/)c0 3629   ~Pcpw 3800    e. cmpt 4267   suc csuc 4584   dom cdm 4879   ` cfv 5455   reccrdg 6668   R1cr1 7689
This theorem is referenced by:  r1suc  7697  r1fin  7700  r1tr  7703  r1ordg  7705  r1pwss  7711  r1val1  7713  rankwflemb  7720  r1elwf  7723  rankr1ai  7725  rankr1bg  7730  pwwf  7734  unwf  7737  uniwf  7746  rankonidlem  7755  rankr1id  7789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-recs 6634  df-rdg 6669  df-r1 7691
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