MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1sucg Structured version   Unicode version

Theorem r1sucg 8088
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 6990 . . 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 8083 . . . 4  |-  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
32dmeqi 5150 . . 3  |-  dom  R1  =  dom  rec ( ( x  e.  _V  |->  ~P x ) ,  (/) )
41, 3eleq2s 2562 . 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 5801 . 2  |-  ( R1
`  suc  A )  =  ( rec (
( x  e.  _V  |->  ~P x ) ,  (/) ) `  suc  A )
6 fvex 5810 . . . 4  |-  ( R1
`  A )  e. 
_V
7 pweq 3972 . . . . 5  |-  ( x  =  ( R1 `  A )  ->  ~P x  =  ~P ( R1 `  A ) )
8 eqid 2454 . . . . 5  |-  ( x  e.  _V  |->  ~P x
)  =  ( x  e.  _V  |->  ~P x
)
96pwex 4584 . . . . 5  |-  ~P ( R1 `  A )  e. 
_V
107, 8, 9fvmpt 5884 . . . 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 5801 . . . 4  |-  ( R1
`  A )  =  ( rec ( ( x  e.  _V  |->  ~P x ) ,  (/) ) `  A )
1312fveq2i 5803 . . 3  |-  ( ( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ( ( x  e.  _V  |->  ~P x
) `  ( rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) ) `  A ) )
1411, 13eqtr3i 2485 . 2  |-  ~P ( R1 `  A )  =  ( ( x  e. 
_V  |->  ~P x ) `  ( rec ( ( x  e.  _V  |->  ~P x
) ,  (/) ) `  A ) )
154, 5, 143eqtr4g 2520 1  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3746   ~Pcpw 3969    |-> cmpt 4459   suc csuc 4830   dom cdm 4949   ` cfv 5527   reccrdg 6976   R1cr1 8081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-recs 6943  df-rdg 6977  df-r1 8083
This theorem is referenced by:  r1suc  8089  r1fin  8092  r1tr  8095  r1ordg  8097  r1pwss  8103  r1val1  8105  rankwflemb  8112  r1elwf  8115  rankr1ai  8117  rankr1bg  8122  pwwf  8126  unwf  8129  uniwf  8138  rankonidlem  8147  rankr1id  8181
  Copyright terms: Public domain W3C validator