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Theorem r1val3 7507
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 7436 . . . . 5  |-  R1  Fn  On
2 fndm 5310 . . . . 5  |-  ( R1  Fn  On  ->  dom  R1  =  On )
31, 2ax-mp 10 . . . 4  |-  dom  R1  =  On
43eleq2i 2350 . . 3  |-  ( A  e.  dom  R1  <->  A  e.  On )
5 r1val1 7455 . . 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 4418 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  A )  ->  x  e.  On )
8 r1val2 7506 . . . . 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 3633 . . 3  |-  ( ( A  e.  On  /\  x  e.  A )  ->  ~P ( R1 `  x )  =  ~P { y  |  (
rank `  y )  e.  x } )
1110iuneq2dv 3929 . 2  |-  ( A  e.  On  ->  U_ x  e.  A  ~P ( R1 `  x )  = 
U_ x  e.  A  ~P { y  |  (
rank `  y )  e.  x } )
126, 11eqtrd 2318 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 1625    e. wcel 1687   {cab 2272   ~Pcpw 3628   U_ciun 3908   Oncon0 4393   dom cdm 4690    Fn wfn 5218   ` cfv 5223   R1cr1 7431   rankcrnk 7432
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-reg 7303  ax-inf2 7339
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rex 2552  df-reu 2553  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-recs 6385  df-rdg 6420  df-r1 7433  df-rank 7434
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