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Theorem r1sucg 8240
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 7141 . . 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 8235 . . . 4  |-  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
32dmeqi 5036 . . 3  |-  dom  R1  =  dom  rec ( ( x  e.  _V  |->  ~P x ) ,  (/) )
41, 3eleq2s 2547 . 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 5866 . 2  |-  ( R1
`  suc  A )  =  ( rec (
( x  e.  _V  |->  ~P x ) ,  (/) ) `  suc  A )
6 fvex 5875 . . . 4  |-  ( R1
`  A )  e. 
_V
7 pweq 3954 . . . . 5  |-  ( x  =  ( R1 `  A )  ->  ~P x  =  ~P ( R1 `  A ) )
8 eqid 2451 . . . . 5  |-  ( x  e.  _V  |->  ~P x
)  =  ( x  e.  _V  |->  ~P x
)
96pwex 4586 . . . . 5  |-  ~P ( R1 `  A )  e. 
_V
107, 8, 9fvmpt 5948 . . . 4  |-  ( ( R1 `  A )  e.  _V  ->  (
( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ~P ( R1
`  A ) )
116, 10ax-mp 5 . . 3  |-  ( ( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ~P ( R1
`  A )
122fveq1i 5866 . . . 4  |-  ( R1
`  A )  =  ( rec ( ( x  e.  _V  |->  ~P x ) ,  (/) ) `  A )
1312fveq2i 5868 . . 3  |-  ( ( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ( ( x  e.  _V  |->  ~P x
) `  ( rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) ) `  A ) )
1411, 13eqtr3i 2475 . 2  |-  ~P ( R1 `  A )  =  ( ( x  e. 
_V  |->  ~P x ) `  ( rec ( ( x  e.  _V  |->  ~P x
) ,  (/) ) `  A ) )
154, 5, 143eqtr4g 2510 1  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444    e. wcel 1887   _Vcvv 3045   (/)c0 3731   ~Pcpw 3951    |-> cmpt 4461   dom cdm 4834   suc csuc 5425   ` cfv 5582   reccrdg 7127   R1cr1 8233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-r1 8235
This theorem is referenced by:  r1suc  8241  r1fin  8244  r1tr  8247  r1ordg  8249  r1pwss  8255  r1val1  8257  rankwflemb  8264  r1elwf  8267  rankr1ai  8269  rankr1bg  8274  pwwf  8278  unwf  8281  uniwf  8290  rankonidlem  8299  rankr1id  8333
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