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Theorem r1val3 7443
Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
r1val3  |-  ( A  e.  On  ->  ( R1 `  A )  = 
U_ x  e.  A  ~P { y  |  (
rank `  y )  e.  x } )
Distinct variable group:    x, y, A

Proof of Theorem r1val3
StepHypRef Expression
1 r1fnon 7372 . . . . 5  |-  R1  Fn  On
2 fndm 5246 . . . . 5  |-  ( R1  Fn  On  ->  dom  R1  =  On )
31, 2ax-mp 10 . . . 4  |-  dom  R1  =  On
43eleq2i 2320 . . 3  |-  ( A  e.  dom  R1  <->  A  e.  On )
5 r1val1 7391 . . 3  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  =  U_ x  e.  A  ~P ( R1
`  x ) )
64, 5sylbir 206 . 2  |-  ( A  e.  On  ->  ( R1 `  A )  = 
U_ x  e.  A  ~P ( R1 `  x
) )
7 onelon 4354 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  A )  ->  x  e.  On )
8 r1val2 7442 . . . . 5  |-  ( x  e.  On  ->  ( R1 `  x )  =  { y  |  (
rank `  y )  e.  x } )
97, 8syl 17 . . . 4  |-  ( ( A  e.  On  /\  x  e.  A )  ->  ( R1 `  x
)  =  { y  |  ( rank `  y
)  e.  x }
)
109pweqd 3571 . . 3  |-  ( ( A  e.  On  /\  x  e.  A )  ->  ~P ( R1 `  x )  =  ~P { y  |  (
rank `  y )  e.  x } )
1110iuneq2dv 3867 . 2  |-  ( A  e.  On  ->  U_ x  e.  A  ~P ( R1 `  x )  = 
U_ x  e.  A  ~P { y  |  (
rank `  y )  e.  x } )
126, 11eqtrd 2288 1  |-  ( A  e.  On  ->  ( R1 `  A )  = 
U_ x  e.  A  ~P { y  |  (
rank `  y )  e.  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2242   ~Pcpw 3566   U_ciun 3846   Oncon0 4329   dom cdm 4626    Fn wfn 4633   ` cfv 4638   R1cr1 7367   rankcrnk 7368
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-reg 7239  ax-inf2 7275
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-recs 6321  df-rdg 6356  df-r1 7369  df-rank 7370
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