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Theorem r1val1 7676
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 7656 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 449 . . . . . . 7  |-  Lim  dom  R1
3 limord 4608 . . . . . . 7  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
42, 3ax-mp 8 . . . . . 6  |-  Ord  dom  R1
5 ordsson 4737 . . . . . 6  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
64, 5ax-mp 8 . . . . 5  |-  dom  R1  C_  On
76sseli 3312 . . . 4  |-  ( A  e.  dom  R1  ->  A  e.  On )
8 onzsl 4793 . . . 4  |-  ( A  e.  On  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A
) ) )
97, 8sylib 189 . . 3  |-  ( A  e.  dom  R1  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
10 simpr 448 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  ->  A  =  (/) )
1110fveq2d 5699 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  =  ( R1
`  (/) ) )
12 r10 7658 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
1311, 12syl6eq 2460 . . . . 5  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  =  (/) )
14 0ss 3624 . . . . . 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 3350 . . . 4  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) )
17 nfv 1626 . . . . . 6  |-  F/ x  A  e.  dom  R1
18 nfcv 2548 . . . . . . 7  |-  F/_ x
( R1 `  A
)
19 nfiu1 4089 . . . . . . 7  |-  F/_ x U_ x  e.  A  ~P ( R1 `  x
)
2018, 19nfss 3309 . . . . . 6  |-  F/ x
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x )
21 simpr 448 . . . . . . . . . . 11  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  A  =  suc  x )
2221fveq2d 5699 . . . . . . . . . 10  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  =  ( R1 `  suc  x
) )
23 eleq1 2472 . . . . . . . . . . . . 13  |-  ( A  =  suc  x  -> 
( A  e.  dom  R1  <->  suc  x  e.  dom  R1 ) )
2423biimpac 473 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  suc  x  e. 
dom  R1 )
25 limsuc 4796 . . . . . . . . . . . . 13  |-  ( Lim 
dom  R1  ->  ( x  e.  dom  R1  <->  suc  x  e. 
dom  R1 ) )
262, 25ax-mp 8 . . . . . . . . . . . 12  |-  ( x  e.  dom  R1  <->  suc  x  e. 
dom  R1 )
2724, 26sylibr 204 . . . . . . . . . . 11  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  x  e.  dom  R1 )
28 r1sucg 7659 . . . . . . . . . . 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 2444 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  =  ~P ( R1 `  x ) )
31 vex 2927 . . . . . . . . . . . 12  |-  x  e. 
_V
3231sucid 4628 . . . . . . . . . . 11  |-  x  e. 
suc  x
3332, 21syl5eleqr 2499 . . . . . . . . . 10  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  x  e.  A )
34 ssiun2 4102 . . . . . . . . . 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 3350 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
3736ex 424 . . . . . . 7  |-  ( A  e.  dom  R1  ->  ( A  =  suc  x  ->  ( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) )
3837a1d 23 . . . . . 6  |-  ( A  e.  dom  R1  ->  ( x  e.  On  ->  ( A  =  suc  x  ->  ( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) ) )
3917, 20, 38rexlimd 2795 . . . . 5  |-  ( A  e.  dom  R1  ->  ( E. x  e.  On  A  =  suc  x  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) )
4039imp 419 . . . 4  |-  ( ( A  e.  dom  R1  /\ 
E. x  e.  On  A  =  suc  x )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
41 r1limg 7661 . . . . . 6  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  =  U_ x  e.  A  ( R1 `  x ) )
42 r1tr 7666 . . . . . . . . . 10  |-  Tr  ( R1 `  x )
43 dftr4 4275 . . . . . . . . . 10  |-  ( Tr  ( R1 `  x
)  <->  ( R1 `  x )  C_  ~P ( R1 `  x ) )
4442, 43mpbi 200 . . . . . . . . 9  |-  ( R1
`  x )  C_  ~P ( R1 `  x
)
4544a1i 11 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  x
)  C_  ~P ( R1 `  x ) )
4645ralrimivw 2758 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  ->  A. x  e.  A  ( R1 `  x ) 
C_  ~P ( R1 `  x ) )
47 ss2iun 4076 . . . . . . 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 3350 . . . . 5  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) )
5049adantrl 697 . . . 4  |-  ( ( A  e.  dom  R1  /\  ( A  e.  _V  /\ 
Lim  A ) )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
5116, 40, 503jaodan 1250 . . 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 650 . 2  |-  ( A  e.  dom  R1  ->  ( R1 `  A ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
53 ordtr1 4592 . . . . . . . 8  |-  ( Ord 
dom  R1  ->  ( ( x  e.  A  /\  A  e.  dom  R1 )  ->  x  e.  dom  R1 ) )
544, 53ax-mp 8 . . . . . . 7  |-  ( ( x  e.  A  /\  A  e.  dom  R1 )  ->  x  e.  dom  R1 )
5554ancoms 440 . . . . . 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 448 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  A
)
58 ordelord 4571 . . . . . . . . . 10  |-  ( ( Ord  dom  R1  /\  A  e.  dom  R1 )  ->  Ord  A )
594, 58mpan 652 . . . . . . . . 9  |-  ( A  e.  dom  R1  ->  Ord 
A )
6059adantr 452 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  Ord  A )
61 ordelsuc 4767 . . . . . . . 8  |-  ( ( x  e.  A  /\  Ord  A )  ->  (
x  e.  A  <->  suc  x  C_  A ) )
6257, 60, 61syl2anc 643 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( x  e.  A  <->  suc  x  C_  A
) )
6357, 62mpbid 202 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  suc  x  C_  A
)
6455, 26sylib 189 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  suc  x  e.  dom  R1 )
65 simpl 444 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  A  e.  dom  R1 )
66 r1ord3g 7669 . . . . . . 7  |-  ( ( suc  x  e.  dom  R1 
/\  A  e.  dom  R1 )  ->  ( suc  x  C_  A  ->  ( R1 `  suc  x ) 
C_  ( R1 `  A ) ) )
6764, 65, 66syl2anc 643 . . . . . 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 3351 . . . 4  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ~P ( R1
`  x )  C_  ( R1 `  A ) )
7069ralrimiva 2757 . . 3  |-  ( A  e.  dom  R1  ->  A. x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A ) )
71 iunss 4100 . . 3  |-  ( U_ x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A )  <->  A. x  e.  A  ~P ( R1 `  x )  C_  ( R1 `  A ) )
7270, 71sylibr 204 . 2  |-  ( A  e.  dom  R1  ->  U_ x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A ) )
7352, 72eqssd 3333 1  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  =  U_ x  e.  A  ~P ( R1
`  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675   _Vcvv 2924    C_ wss 3288   (/)c0 3596   ~Pcpw 3767   U_ciun 4061   Tr wtr 4270   Ord word 4548   Oncon0 4549   Lim wlim 4550   suc csuc 4551   dom cdm 4845   Fun wfun 5415   ` cfv 5421   R1cr1 7652
This theorem is referenced by:  rankr1ai  7688  r1val3  7728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-recs 6600  df-rdg 6635  df-r1 7654
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