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Theorem r1sucg 7325
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
StepHypRef Expression
1 rdgsucg 6322 . . 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 7320 . . . 4  |-  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
32dmeqi 4787 . . 3  |-  dom  R1  =  dom  rec ( ( x  e.  _V  |->  ~P x ) ,  (/) )
41, 3eleq2s 2345 . 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 5378 . 2  |-  ( R1
`  suc  A )  =  ( rec (
( x  e.  _V  |->  ~P x ) ,  (/) ) `  suc  A )
6 fvex 5391 . . . 4  |-  ( R1
`  A )  e. 
_V
7 pweq 3533 . . . . 5  |-  ( x  =  ( R1 `  A )  ->  ~P x  =  ~P ( R1 `  A ) )
8 eqid 2253 . . . . 5  |-  ( x  e.  _V  |->  ~P x
)  =  ( x  e.  _V  |->  ~P x
)
96pwex 4087 . . . . 5  |-  ~P ( R1 `  A )  e. 
_V
107, 8, 9fvmpt 5454 . . . 4  |-  ( ( R1 `  A )  e.  _V  ->  (
( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ~P ( R1
`  A ) )
116, 10ax-mp 10 . . 3  |-  ( ( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ~P ( R1
`  A )
122fveq1i 5378 . . . 4  |-  ( R1
`  A )  =  ( rec ( ( x  e.  _V  |->  ~P x ) ,  (/) ) `  A )
1312fveq2i 5380 . . 3  |-  ( ( x  e.  _V  |->  ~P x ) `  ( R1 `  A ) )  =  ( ( x  e.  _V  |->  ~P x
) `  ( rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) ) `  A ) )
1411, 13eqtr3i 2275 . 2  |-  ~P ( R1 `  A )  =  ( ( x  e. 
_V  |->  ~P x ) `  ( rec ( ( x  e.  _V  |->  ~P x
) ,  (/) ) `  A ) )
154, 5, 143eqtr4g 2310 1  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   _Vcvv 2727   (/)c0 3362   ~Pcpw 3530    e. cmpt 3974   suc csuc 4287   dom cdm 4580   ` cfv 4592   reccrdg 6308   R1cr1 7318
This theorem is referenced by:  r1suc  7326  r1fin  7329  r1tr  7332  r1ordg  7334  r1pwss  7340  r1val1  7342  rankwflemb  7349  r1elwf  7352  rankr1ai  7354  rankr1bg  7359  pwwf  7363  unwf  7366  uniwf  7375  rankonidlem  7384  rankr1id  7418
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-recs 6274  df-rdg 6309  df-r1 7320
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