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Theorem r1sucg 7443
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 6438 . . 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 7438 . . . 4  |-  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
32dmeqi 4882 . . 3  |-  dom  R1  =  dom  rec ( ( x  e.  _V  |->  ~P x ) ,  (/) )
41, 3eleq2s 2377 . 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 5528 . 2  |-  ( R1
`  suc  A )  =  ( rec (
( x  e.  _V  |->  ~P x ) ,  (/) ) `  suc  A )
6 fvex 5541 . . . 4  |-  ( R1
`  A )  e. 
_V
7 pweq 3630 . . . . 5  |-  ( x  =  ( R1 `  A )  ->  ~P x  =  ~P ( R1 `  A ) )
8 eqid 2285 . . . . 5  |-  ( x  e.  _V  |->  ~P x
)  =  ( x  e.  _V  |->  ~P x
)
96pwex 4195 . . . . 5  |-  ~P ( R1 `  A )  e. 
_V
107, 8, 9fvmpt 5604 . . . 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 5528 . . . 4  |-  ( R1
`  A )  =  ( rec ( ( x  e.  _V  |->  ~P x ) ,  (/) ) `  A )
1312fveq2i 5530 . . 3  |-  ( ( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ( ( x  e.  _V  |->  ~P x
) `  ( rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) ) `  A ) )
1411, 13eqtr3i 2307 . 2  |-  ~P ( R1 `  A )  =  ( ( x  e. 
_V  |->  ~P x ) `  ( rec ( ( x  e.  _V  |->  ~P x
) ,  (/) ) `  A ) )
154, 5, 143eqtr4g 2342 1  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    e. wcel 1686   _Vcvv 2790   (/)c0 3457   ~Pcpw 3627    e. cmpt 4079   suc csuc 4396   dom cdm 4691   ` cfv 5257   reccrdg 6424   R1cr1 7436
This theorem is referenced by:  r1suc  7444  r1fin  7447  r1tr  7450  r1ordg  7452  r1pwss  7458  r1val1  7460  rankwflemb  7467  r1elwf  7470  rankr1ai  7472  rankr1bg  7477  pwwf  7481  unwf  7484  uniwf  7493  rankonidlem  7502  rankr1id  7536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425  df-r1 7438
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