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Theorem r1val1 7989
Description: The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
r1val1  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  =  U_ x  e.  A  ~P ( R1
`  x ) )
Distinct variable group:    x, A

Proof of Theorem r1val1
StepHypRef Expression
1 r1funlim 7969 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 459 . . . . . . 7  |-  Lim  dom  R1
3 limord 4774 . . . . . . 7  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
42, 3ax-mp 5 . . . . . 6  |-  Ord  dom  R1
5 ordsson 6400 . . . . . 6  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
64, 5ax-mp 5 . . . . 5  |-  dom  R1  C_  On
76sseli 3349 . . . 4  |-  ( A  e.  dom  R1  ->  A  e.  On )
8 onzsl 6456 . . . 4  |-  ( A  e.  On  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A
) ) )
97, 8sylib 196 . . 3  |-  ( A  e.  dom  R1  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
10 simpr 458 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  ->  A  =  (/) )
1110fveq2d 5692 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  =  ( R1
`  (/) ) )
12 r10 7971 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
1311, 12syl6eq 2489 . . . . 5  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  =  (/) )
14 0ss 3663 . . . . . 6  |-  (/)  C_  U_ x  e.  A  ~P ( R1 `  x )
1514a1i 11 . . . . 5  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  ->  (/)  C_  U_ x  e.  A  ~P ( R1 `  x
) )
1613, 15eqsstrd 3387 . . . 4  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) )
17 nfv 1678 . . . . . 6  |-  F/ x  A  e.  dom  R1
18 nfcv 2577 . . . . . . 7  |-  F/_ x
( R1 `  A
)
19 nfiu1 4197 . . . . . . 7  |-  F/_ x U_ x  e.  A  ~P ( R1 `  x
)
2018, 19nfss 3346 . . . . . 6  |-  F/ x
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x )
21 simpr 458 . . . . . . . . . . 11  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  A  =  suc  x )
2221fveq2d 5692 . . . . . . . . . 10  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  =  ( R1 `  suc  x
) )
23 eleq1 2501 . . . . . . . . . . . . 13  |-  ( A  =  suc  x  -> 
( A  e.  dom  R1  <->  suc  x  e.  dom  R1 ) )
2423biimpac 483 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  suc  x  e. 
dom  R1 )
25 limsuc 6459 . . . . . . . . . . . . 13  |-  ( Lim 
dom  R1  ->  ( x  e.  dom  R1  <->  suc  x  e. 
dom  R1 ) )
262, 25ax-mp 5 . . . . . . . . . . . 12  |-  ( x  e.  dom  R1  <->  suc  x  e. 
dom  R1 )
2724, 26sylibr 212 . . . . . . . . . . 11  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  x  e.  dom  R1 )
28 r1sucg 7972 . . . . . . . . . . 11  |-  ( x  e.  dom  R1  ->  ( R1 `  suc  x
)  =  ~P ( R1 `  x ) )
2927, 28syl 16 . . . . . . . . . 10  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 ` 
suc  x )  =  ~P ( R1 `  x ) )
3022, 29eqtrd 2473 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  =  ~P ( R1 `  x ) )
31 vex 2973 . . . . . . . . . . . 12  |-  x  e. 
_V
3231sucid 4794 . . . . . . . . . . 11  |-  x  e. 
suc  x
3332, 21syl5eleqr 2528 . . . . . . . . . 10  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  x  e.  A )
34 ssiun2 4210 . . . . . . . . . 10  |-  ( x  e.  A  ->  ~P ( R1 `  x ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
3533, 34syl 16 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ~P ( R1 `  x )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
3630, 35eqsstrd 3387 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
3736ex 434 . . . . . . 7  |-  ( A  e.  dom  R1  ->  ( A  =  suc  x  ->  ( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) )
3837a1d 25 . . . . . 6  |-  ( A  e.  dom  R1  ->  ( x  e.  On  ->  ( A  =  suc  x  ->  ( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) ) )
3917, 20, 38rexlimd 2836 . . . . 5  |-  ( A  e.  dom  R1  ->  ( E. x  e.  On  A  =  suc  x  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) )
4039imp 429 . . . 4  |-  ( ( A  e.  dom  R1  /\ 
E. x  e.  On  A  =  suc  x )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
41 r1limg 7974 . . . . . 6  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  =  U_ x  e.  A  ( R1 `  x ) )
42 r1tr 7979 . . . . . . . . . 10  |-  Tr  ( R1 `  x )
43 dftr4 4387 . . . . . . . . . 10  |-  ( Tr  ( R1 `  x
)  <->  ( R1 `  x )  C_  ~P ( R1 `  x ) )
4442, 43mpbi 208 . . . . . . . . 9  |-  ( R1
`  x )  C_  ~P ( R1 `  x
)
4544a1i 11 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  x
)  C_  ~P ( R1 `  x ) )
4645ralrimivw 2798 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  ->  A. x  e.  A  ( R1 `  x ) 
C_  ~P ( R1 `  x ) )
47 ss2iun 4183 . . . . . . 7  |-  ( A. x  e.  A  ( R1 `  x )  C_  ~P ( R1 `  x
)  ->  U_ x  e.  A  ( R1 `  x )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
4846, 47syl 16 . . . . . 6  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  ->  U_ x  e.  A  ( R1 `  x ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
4941, 48eqsstrd 3387 . . . . 5  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) )
5049adantrl 710 . . . 4  |-  ( ( A  e.  dom  R1  /\  ( A  e.  _V  /\ 
Lim  A ) )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
5116, 40, 503jaodan 1279 . . 3  |-  ( ( A  e.  dom  R1  /\  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\ 
Lim  A ) ) )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
529, 51mpdan 663 . 2  |-  ( A  e.  dom  R1  ->  ( R1 `  A ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
53 ordtr1 4758 . . . . . . . 8  |-  ( Ord 
dom  R1  ->  ( ( x  e.  A  /\  A  e.  dom  R1 )  ->  x  e.  dom  R1 ) )
544, 53ax-mp 5 . . . . . . 7  |-  ( ( x  e.  A  /\  A  e.  dom  R1 )  ->  x  e.  dom  R1 )
5554ancoms 450 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  dom  R1 )
5655, 28syl 16 . . . . 5  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( R1 `  suc  x )  =  ~P ( R1 `  x ) )
57 simpr 458 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  A
)
58 ordelord 4737 . . . . . . . . . 10  |-  ( ( Ord  dom  R1  /\  A  e.  dom  R1 )  ->  Ord  A )
594, 58mpan 665 . . . . . . . . 9  |-  ( A  e.  dom  R1  ->  Ord 
A )
6059adantr 462 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  Ord  A )
61 ordelsuc 6430 . . . . . . . 8  |-  ( ( x  e.  A  /\  Ord  A )  ->  (
x  e.  A  <->  suc  x  C_  A ) )
6257, 60, 61syl2anc 656 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( x  e.  A  <->  suc  x  C_  A
) )
6357, 62mpbid 210 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  suc  x  C_  A
)
6455, 26sylib 196 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  suc  x  e.  dom  R1 )
65 simpl 454 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  A  e.  dom  R1 )
66 r1ord3g 7982 . . . . . . 7  |-  ( ( suc  x  e.  dom  R1 
/\  A  e.  dom  R1 )  ->  ( suc  x  C_  A  ->  ( R1 `  suc  x ) 
C_  ( R1 `  A ) ) )
6764, 65, 66syl2anc 656 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( suc  x  C_  A  ->  ( R1 ` 
suc  x )  C_  ( R1 `  A ) ) )
6863, 67mpd 15 . . . . 5  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( R1 `  suc  x )  C_  ( R1 `  A ) )
6956, 68eqsstr3d 3388 . . . 4  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ~P ( R1
`  x )  C_  ( R1 `  A ) )
7069ralrimiva 2797 . . 3  |-  ( A  e.  dom  R1  ->  A. x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A ) )
71 iunss 4208 . . 3  |-  ( U_ x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A )  <->  A. x  e.  A  ~P ( R1 `  x )  C_  ( R1 `  A ) )
7270, 71sylibr 212 . 2  |-  ( A  e.  dom  R1  ->  U_ x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A ) )
7352, 72eqssd 3370 1  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  =  U_ x  e.  A  ~P ( R1
`  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 959    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   _Vcvv 2970    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   U_ciun 4168   Tr wtr 4382   Ord word 4714   Oncon0 4715   Lim wlim 4716   suc csuc 4717   dom cdm 4836   Fun wfun 5409   ` cfv 5415   R1cr1 7965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-recs 6828  df-rdg 6862  df-r1 7967
This theorem is referenced by:  rankr1ai  8001  r1val3  8041
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