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Theorem r1val1 8117
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 simpr 459 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  ->  A  =  (/) )
21fveq2d 5778 . . . . 5  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  =  ( R1
`  (/) ) )
3 r10 8099 . . . . 5  |-  ( R1
`  (/) )  =  (/)
42, 3syl6eq 2439 . . . 4  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  =  (/) )
5 0ss 3741 . . . . 5  |-  (/)  C_  U_ x  e.  A  ~P ( R1 `  x )
65a1i 11 . . . 4  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  ->  (/)  C_  U_ x  e.  A  ~P ( R1 `  x
) )
74, 6eqsstrd 3451 . . 3  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) )
8 nfv 1715 . . . . 5  |-  F/ x  A  e.  dom  R1
9 nfcv 2544 . . . . . 6  |-  F/_ x
( R1 `  A
)
10 nfiu1 4273 . . . . . 6  |-  F/_ x U_ x  e.  A  ~P ( R1 `  x
)
119, 10nfss 3410 . . . . 5  |-  F/ x
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x )
12 simpr 459 . . . . . . . . . 10  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  A  =  suc  x )
1312fveq2d 5778 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  =  ( R1 `  suc  x
) )
14 eleq1 2454 . . . . . . . . . . . 12  |-  ( A  =  suc  x  -> 
( A  e.  dom  R1  <->  suc  x  e.  dom  R1 ) )
1514biimpac 484 . . . . . . . . . . 11  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  suc  x  e. 
dom  R1 )
16 r1funlim 8097 . . . . . . . . . . . . 13  |-  ( Fun 
R1  /\  Lim  dom  R1 )
1716simpri 460 . . . . . . . . . . . 12  |-  Lim  dom  R1
18 limsuc 6583 . . . . . . . . . . . 12  |-  ( Lim 
dom  R1  ->  ( x  e.  dom  R1  <->  suc  x  e. 
dom  R1 ) )
1917, 18ax-mp 5 . . . . . . . . . . 11  |-  ( x  e.  dom  R1  <->  suc  x  e. 
dom  R1 )
2015, 19sylibr 212 . . . . . . . . . 10  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  x  e.  dom  R1 )
21 r1sucg 8100 . . . . . . . . . 10  |-  ( x  e.  dom  R1  ->  ( R1 `  suc  x
)  =  ~P ( R1 `  x ) )
2220, 21syl 16 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 ` 
suc  x )  =  ~P ( R1 `  x ) )
2313, 22eqtrd 2423 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  =  ~P ( R1 `  x ) )
24 vex 3037 . . . . . . . . . . 11  |-  x  e. 
_V
2524sucid 4871 . . . . . . . . . 10  |-  x  e. 
suc  x
2625, 12syl5eleqr 2477 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  x  e.  A )
27 ssiun2 4286 . . . . . . . . 9  |-  ( x  e.  A  ->  ~P ( R1 `  x ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
2826, 27syl 16 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ~P ( R1 `  x )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
2923, 28eqsstrd 3451 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
3029ex 432 . . . . . 6  |-  ( A  e.  dom  R1  ->  ( A  =  suc  x  ->  ( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) )
3130a1d 25 . . . . 5  |-  ( A  e.  dom  R1  ->  ( x  e.  On  ->  ( A  =  suc  x  ->  ( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) ) )
328, 11, 31rexlimd 2866 . . . 4  |-  ( A  e.  dom  R1  ->  ( E. x  e.  On  A  =  suc  x  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) )
3332imp 427 . . 3  |-  ( ( A  e.  dom  R1  /\ 
E. x  e.  On  A  =  suc  x )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
34 r1limg 8102 . . . . 5  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  =  U_ x  e.  A  ( R1 `  x ) )
35 r1tr 8107 . . . . . . . . 9  |-  Tr  ( R1 `  x )
36 dftr4 4465 . . . . . . . . 9  |-  ( Tr  ( R1 `  x
)  <->  ( R1 `  x )  C_  ~P ( R1 `  x ) )
3735, 36mpbi 208 . . . . . . . 8  |-  ( R1
`  x )  C_  ~P ( R1 `  x
)
3837a1i 11 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  x
)  C_  ~P ( R1 `  x ) )
3938ralrimivw 2797 . . . . . 6  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  ->  A. x  e.  A  ( R1 `  x ) 
C_  ~P ( R1 `  x ) )
40 ss2iun 4259 . . . . . 6  |-  ( A. x  e.  A  ( R1 `  x )  C_  ~P ( R1 `  x
)  ->  U_ x  e.  A  ( R1 `  x )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
4139, 40syl 16 . . . . 5  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  ->  U_ x  e.  A  ( R1 `  x ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
4234, 41eqsstrd 3451 . . . 4  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) )
4342adantrl 713 . . 3  |-  ( ( A  e.  dom  R1  /\  ( A  e.  _V  /\ 
Lim  A ) )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
44 limord 4851 . . . . . . 7  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4517, 44ax-mp 5 . . . . . 6  |-  Ord  dom  R1
46 ordsson 6524 . . . . . 6  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
4745, 46ax-mp 5 . . . . 5  |-  dom  R1  C_  On
4847sseli 3413 . . . 4  |-  ( A  e.  dom  R1  ->  A  e.  On )
49 onzsl 6580 . . . 4  |-  ( A  e.  On  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A
) ) )
5048, 49sylib 196 . . 3  |-  ( A  e.  dom  R1  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
517, 33, 43, 50mpjao3dan 1293 . 2  |-  ( A  e.  dom  R1  ->  ( R1 `  A ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
52 ordtr1 4835 . . . . . . . 8  |-  ( Ord 
dom  R1  ->  ( ( x  e.  A  /\  A  e.  dom  R1 )  ->  x  e.  dom  R1 ) )
5345, 52ax-mp 5 . . . . . . 7  |-  ( ( x  e.  A  /\  A  e.  dom  R1 )  ->  x  e.  dom  R1 )
5453ancoms 451 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  dom  R1 )
5554, 21syl 16 . . . . 5  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( R1 `  suc  x )  =  ~P ( R1 `  x ) )
56 simpr 459 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  A
)
57 ordelord 4814 . . . . . . . . . 10  |-  ( ( Ord  dom  R1  /\  A  e.  dom  R1 )  ->  Ord  A )
5845, 57mpan 668 . . . . . . . . 9  |-  ( A  e.  dom  R1  ->  Ord 
A )
5958adantr 463 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  Ord  A )
60 ordelsuc 6554 . . . . . . . 8  |-  ( ( x  e.  A  /\  Ord  A )  ->  (
x  e.  A  <->  suc  x  C_  A ) )
6156, 59, 60syl2anc 659 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( x  e.  A  <->  suc  x  C_  A
) )
6256, 61mpbid 210 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  suc  x  C_  A
)
6354, 19sylib 196 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  suc  x  e.  dom  R1 )
64 simpl 455 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  A  e.  dom  R1 )
65 r1ord3g 8110 . . . . . . 7  |-  ( ( suc  x  e.  dom  R1 
/\  A  e.  dom  R1 )  ->  ( suc  x  C_  A  ->  ( R1 `  suc  x ) 
C_  ( R1 `  A ) ) )
6663, 64, 65syl2anc 659 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( suc  x  C_  A  ->  ( R1 ` 
suc  x )  C_  ( R1 `  A ) ) )
6762, 66mpd 15 . . . . 5  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( R1 `  suc  x )  C_  ( R1 `  A ) )
6855, 67eqsstr3d 3452 . . . 4  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ~P ( R1
`  x )  C_  ( R1 `  A ) )
6968ralrimiva 2796 . . 3  |-  ( A  e.  dom  R1  ->  A. x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A ) )
70 iunss 4284 . . 3  |-  ( U_ x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A )  <->  A. x  e.  A  ~P ( R1 `  x )  C_  ( R1 `  A ) )
7169, 70sylibr 212 . 2  |-  ( A  e.  dom  R1  ->  U_ x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A ) )
7251, 71eqssd 3434 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 367    \/ w3o 970    = wceq 1399    e. wcel 1826   A.wral 2732   E.wrex 2733   _Vcvv 3034    C_ wss 3389   (/)c0 3711   ~Pcpw 3927   U_ciun 4243   Tr wtr 4460   Ord word 4791   Oncon0 4792   Lim wlim 4793   suc csuc 4794   dom cdm 4913   Fun wfun 5490   ` cfv 5496   R1cr1 8093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-om 6600  df-recs 6960  df-rdg 6994  df-r1 8095
This theorem is referenced by:  rankr1ai  8129  r1val3  8169
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