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Theorem r1val1 8202
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 461 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  ->  A  =  (/) )
21fveq2d 5856 . . . . 5  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  =  ( R1
`  (/) ) )
3 r10 8184 . . . . 5  |-  ( R1
`  (/) )  =  (/)
42, 3syl6eq 2498 . . . 4  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  =  (/) )
5 0ss 3796 . . . . 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 3520 . . 3  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) )
8 nfv 1692 . . . . 5  |-  F/ x  A  e.  dom  R1
9 nfcv 2603 . . . . . 6  |-  F/_ x
( R1 `  A
)
10 nfiu1 4341 . . . . . 6  |-  F/_ x U_ x  e.  A  ~P ( R1 `  x
)
119, 10nfss 3479 . . . . 5  |-  F/ x
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x )
12 simpr 461 . . . . . . . . . 10  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  A  =  suc  x )
1312fveq2d 5856 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  =  ( R1 `  suc  x
) )
14 eleq1 2513 . . . . . . . . . . . 12  |-  ( A  =  suc  x  -> 
( A  e.  dom  R1  <->  suc  x  e.  dom  R1 ) )
1514biimpac 486 . . . . . . . . . . 11  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  suc  x  e. 
dom  R1 )
16 r1funlim 8182 . . . . . . . . . . . . 13  |-  ( Fun 
R1  /\  Lim  dom  R1 )
1716simpri 462 . . . . . . . . . . . 12  |-  Lim  dom  R1
18 limsuc 6665 . . . . . . . . . . . 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 8185 . . . . . . . . . 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 2482 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  =  ~P ( R1 `  x ) )
24 vex 3096 . . . . . . . . . . 11  |-  x  e. 
_V
2524sucid 4943 . . . . . . . . . 10  |-  x  e. 
suc  x
2625, 12syl5eleqr 2536 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  x  e.  A )
27 ssiun2 4354 . . . . . . . . 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 3520 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
3029ex 434 . . . . . 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 2925 . . . 4  |-  ( A  e.  dom  R1  ->  ( E. x  e.  On  A  =  suc  x  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) )
3332imp 429 . . 3  |-  ( ( A  e.  dom  R1  /\ 
E. x  e.  On  A  =  suc  x )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
34 r1limg 8187 . . . . 5  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  =  U_ x  e.  A  ( R1 `  x ) )
35 r1tr 8192 . . . . . . . . 9  |-  Tr  ( R1 `  x )
36 dftr4 4531 . . . . . . . . 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 2856 . . . . . 6  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  ->  A. x  e.  A  ( R1 `  x ) 
C_  ~P ( R1 `  x ) )
40 ss2iun 4327 . . . . . 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 3520 . . . 4  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) )
4342adantrl 715 . . 3  |-  ( ( A  e.  dom  R1  /\  ( A  e.  _V  /\ 
Lim  A ) )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
44 limord 4923 . . . . . . 7  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4517, 44ax-mp 5 . . . . . 6  |-  Ord  dom  R1
46 ordsson 6606 . . . . . 6  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
4745, 46ax-mp 5 . . . . 5  |-  dom  R1  C_  On
4847sseli 3482 . . . 4  |-  ( A  e.  dom  R1  ->  A  e.  On )
49 onzsl 6662 . . . 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 1294 . 2  |-  ( A  e.  dom  R1  ->  ( R1 `  A ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
52 ordtr1 4907 . . . . . . . 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 453 . . . . . 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 461 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  A
)
57 ordelord 4886 . . . . . . . . . 10  |-  ( ( Ord  dom  R1  /\  A  e.  dom  R1 )  ->  Ord  A )
5845, 57mpan 670 . . . . . . . . 9  |-  ( A  e.  dom  R1  ->  Ord 
A )
5958adantr 465 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  Ord  A )
60 ordelsuc 6636 . . . . . . . 8  |-  ( ( x  e.  A  /\  Ord  A )  ->  (
x  e.  A  <->  suc  x  C_  A ) )
6156, 59, 60syl2anc 661 . . . . . . 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 457 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  A  e.  dom  R1 )
65 r1ord3g 8195 . . . . . . 7  |-  ( ( suc  x  e.  dom  R1 
/\  A  e.  dom  R1 )  ->  ( suc  x  C_  A  ->  ( R1 `  suc  x ) 
C_  ( R1 `  A ) ) )
6663, 64, 65syl2anc 661 . . . . . 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 3521 . . . 4  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ~P ( R1
`  x )  C_  ( R1 `  A ) )
6968ralrimiva 2855 . . 3  |-  ( A  e.  dom  R1  ->  A. x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A ) )
70 iunss 4352 . . 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 3503 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 971    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792   _Vcvv 3093    C_ wss 3458   (/)c0 3767   ~Pcpw 3993   U_ciun 4311   Tr wtr 4526   Ord word 4863   Oncon0 4864   Lim wlim 4865   suc csuc 4866   dom cdm 4985   Fun wfun 5568   ` cfv 5574   R1cr1 8178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-recs 7040  df-rdg 7074  df-r1 8180
This theorem is referenced by:  rankr1ai  8214  r1val3  8254
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