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Theorem r1val3 7510
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 7439 . . . . 5  |-  R1  Fn  On
2 fndm 5343 . . . . 5  |-  ( R1  Fn  On  ->  dom  R1  =  On )
31, 2ax-mp 8 . . . 4  |-  dom  R1  =  On
43eleq2i 2347 . . 3  |-  ( A  e.  dom  R1  <->  A  e.  On )
5 r1val1 7458 . . 3  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  =  U_ x  e.  A  ~P ( R1
`  x ) )
64, 5sylbir 204 . 2  |-  ( A  e.  On  ->  ( R1 `  A )  = 
U_ x  e.  A  ~P ( R1 `  x
) )
7 onelon 4417 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  A )  ->  x  e.  On )
8 r1val2 7509 . . . . 5  |-  ( x  e.  On  ->  ( R1 `  x )  =  { y  |  (
rank `  y )  e.  x } )
97, 8syl 15 . . . 4  |-  ( ( A  e.  On  /\  x  e.  A )  ->  ( R1 `  x
)  =  { y  |  ( rank `  y
)  e.  x }
)
109pweqd 3630 . . 3  |-  ( ( A  e.  On  /\  x  e.  A )  ->  ~P ( R1 `  x )  =  ~P { y  |  (
rank `  y )  e.  x } )
1110iuneq2dv 3926 . 2  |-  ( A  e.  On  ->  U_ x  e.  A  ~P ( R1 `  x )  = 
U_ x  e.  A  ~P { y  |  (
rank `  y )  e.  x } )
126, 11eqtrd 2315 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 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   ~Pcpw 3625   U_ciun 3905   Oncon0 4392   dom cdm 4689    Fn wfn 5250   ` cfv 5255   R1cr1 7434   rankcrnk 7435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
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