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Theorem r1val1 8265
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 462 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  ->  A  =  (/) )
21fveq2d 5885 . . . . 5  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  =  ( R1
`  (/) ) )
3 r10 8247 . . . . 5  |-  ( R1
`  (/) )  =  (/)
42, 3syl6eq 2479 . . . 4  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  =  (/) )
5 0ss 3793 . . . . 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 3498 . . 3  |-  ( ( A  e.  dom  R1  /\  A  =  (/) )  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) )
8 nfv 1755 . . . . 5  |-  F/ x  A  e.  dom  R1
9 nfcv 2580 . . . . . 6  |-  F/_ x
( R1 `  A
)
10 nfiu1 4329 . . . . . 6  |-  F/_ x U_ x  e.  A  ~P ( R1 `  x
)
119, 10nfss 3457 . . . . 5  |-  F/ x
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x )
12 simpr 462 . . . . . . . . . 10  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  A  =  suc  x )
1312fveq2d 5885 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  =  ( R1 `  suc  x
) )
14 eleq1 2495 . . . . . . . . . . . 12  |-  ( A  =  suc  x  -> 
( A  e.  dom  R1  <->  suc  x  e.  dom  R1 ) )
1514biimpac 488 . . . . . . . . . . 11  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  suc  x  e. 
dom  R1 )
16 r1funlim 8245 . . . . . . . . . . . . 13  |-  ( Fun 
R1  /\  Lim  dom  R1 )
1716simpri 463 . . . . . . . . . . . 12  |-  Lim  dom  R1
18 limsuc 6690 . . . . . . . . . . . 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 215 . . . . . . . . . 10  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  x  e.  dom  R1 )
21 r1sucg 8248 . . . . . . . . . 10  |-  ( x  e.  dom  R1  ->  ( R1 `  suc  x
)  =  ~P ( R1 `  x ) )
2220, 21syl 17 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 ` 
suc  x )  =  ~P ( R1 `  x ) )
2313, 22eqtrd 2463 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  =  ~P ( R1 `  x ) )
24 vex 3083 . . . . . . . . . . 11  |-  x  e. 
_V
2524sucid 5521 . . . . . . . . . 10  |-  x  e. 
suc  x
2625, 12syl5eleqr 2514 . . . . . . . . 9  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  x  e.  A )
27 ssiun2 4342 . . . . . . . . 9  |-  ( x  e.  A  ->  ~P ( R1 `  x ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
2826, 27syl 17 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ~P ( R1 `  x )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
2923, 28eqsstrd 3498 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  A  =  suc  x
)  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
3029ex 435 . . . . . 6  |-  ( A  e.  dom  R1  ->  ( A  =  suc  x  ->  ( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) )
3130a1d 26 . . . . 5  |-  ( A  e.  dom  R1  ->  ( x  e.  On  ->  ( A  =  suc  x  ->  ( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) ) )
328, 11, 31rexlimd 2906 . . . 4  |-  ( A  e.  dom  R1  ->  ( E. x  e.  On  A  =  suc  x  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) ) )
3332imp 430 . . 3  |-  ( ( A  e.  dom  R1  /\ 
E. x  e.  On  A  =  suc  x )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
34 r1limg 8250 . . . . 5  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  =  U_ x  e.  A  ( R1 `  x ) )
35 r1tr 8255 . . . . . . . . 9  |-  Tr  ( R1 `  x )
36 dftr4 4523 . . . . . . . . 9  |-  ( Tr  ( R1 `  x
)  <->  ( R1 `  x )  C_  ~P ( R1 `  x ) )
3735, 36mpbi 211 . . . . . . . 8  |-  ( R1
`  x )  C_  ~P ( R1 `  x
)
3837a1i 11 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  x
)  C_  ~P ( R1 `  x ) )
3938ralrimivw 2837 . . . . . 6  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  ->  A. x  e.  A  ( R1 `  x ) 
C_  ~P ( R1 `  x ) )
40 ss2iun 4315 . . . . . 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 17 . . . . 5  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  ->  U_ x  e.  A  ( R1 `  x ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
4234, 41eqsstrd 3498 . . . 4  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  C_  U_ x  e.  A  ~P ( R1
`  x ) )
4342adantrl 720 . . 3  |-  ( ( A  e.  dom  R1  /\  ( A  e.  _V  /\ 
Lim  A ) )  ->  ( R1 `  A )  C_  U_ x  e.  A  ~P ( R1 `  x ) )
44 limord 5501 . . . . . . 7  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4517, 44ax-mp 5 . . . . . 6  |-  Ord  dom  R1
46 ordsson 6630 . . . . . 6  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
4745, 46ax-mp 5 . . . . 5  |-  dom  R1  C_  On
4847sseli 3460 . . . 4  |-  ( A  e.  dom  R1  ->  A  e.  On )
49 onzsl 6687 . . . 4  |-  ( A  e.  On  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A
) ) )
5048, 49sylib 199 . . 3  |-  ( A  e.  dom  R1  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
517, 33, 43, 50mpjao3dan 1331 . 2  |-  ( A  e.  dom  R1  ->  ( R1 `  A ) 
C_  U_ x  e.  A  ~P ( R1 `  x
) )
52 ordtr1 5485 . . . . . . . 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 454 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  dom  R1 )
5554, 21syl 17 . . . . 5  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( R1 `  suc  x )  =  ~P ( R1 `  x ) )
56 simpr 462 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  A
)
57 ordelord 5464 . . . . . . . . . 10  |-  ( ( Ord  dom  R1  /\  A  e.  dom  R1 )  ->  Ord  A )
5845, 57mpan 674 . . . . . . . . 9  |-  ( A  e.  dom  R1  ->  Ord 
A )
5958adantr 466 . . . . . . . 8  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  Ord  A )
60 ordelsuc 6661 . . . . . . . 8  |-  ( ( x  e.  A  /\  Ord  A )  ->  (
x  e.  A  <->  suc  x  C_  A ) )
6156, 59, 60syl2anc 665 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( x  e.  A  <->  suc  x  C_  A
) )
6256, 61mpbid 213 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  suc  x  C_  A
)
6354, 19sylib 199 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  suc  x  e.  dom  R1 )
64 simpl 458 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  A  e.  dom  R1 )
65 r1ord3g 8258 . . . . . . 7  |-  ( ( suc  x  e.  dom  R1 
/\  A  e.  dom  R1 )  ->  ( suc  x  C_  A  ->  ( R1 `  suc  x ) 
C_  ( R1 `  A ) ) )
6663, 64, 65syl2anc 665 . . . . . 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 3499 . . . 4  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ~P ( R1
`  x )  C_  ( R1 `  A ) )
6968ralrimiva 2836 . . 3  |-  ( A  e.  dom  R1  ->  A. x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A ) )
70 iunss 4340 . . 3  |-  ( U_ x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A )  <->  A. x  e.  A  ~P ( R1 `  x )  C_  ( R1 `  A ) )
7169, 70sylibr 215 . 2  |-  ( A  e.  dom  R1  ->  U_ x  e.  A  ~P ( R1 `  x ) 
C_  ( R1 `  A ) )
7251, 71eqssd 3481 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 187    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772   _Vcvv 3080    C_ wss 3436   (/)c0 3761   ~Pcpw 3981   U_ciun 4299   Tr wtr 4518   dom cdm 4853   Ord word 5441   Oncon0 5442   Lim wlim 5443   suc csuc 5444   Fun wfun 5595   ` cfv 5601   R1cr1 8241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-r1 8243
This theorem is referenced by:  rankr1ai  8277  r1val3  8317
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