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Theorem r1val3 8334
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 8263 . . . . 5  |-  R1  Fn  On
2 fndm 5696 . . . . 5  |-  ( R1  Fn  On  ->  dom  R1  =  On )
31, 2ax-mp 5 . . . 4  |-  dom  R1  =  On
43eleq2i 2531 . . 3  |-  ( A  e.  dom  R1  <->  A  e.  On )
5 r1val1 8282 . . 3  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  =  U_ x  e.  A  ~P ( R1
`  x ) )
64, 5sylbir 218 . 2  |-  ( A  e.  On  ->  ( R1 `  A )  = 
U_ x  e.  A  ~P ( R1 `  x
) )
7 onelon 5466 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  A )  ->  x  e.  On )
8 r1val2 8333 . . . . 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 3967 . . 3  |-  ( ( A  e.  On  /\  x  e.  A )  ->  ~P ( R1 `  x )  =  ~P { y  |  (
rank `  y )  e.  x } )
1110iuneq2dv 4313 . 2  |-  ( A  e.  On  ->  U_ x  e.  A  ~P ( R1 `  x )  = 
U_ x  e.  A  ~P { y  |  (
rank `  y )  e.  x } )
126, 11eqtrd 2495 1  |-  ( A  e.  On  ->  ( R1 `  A )  = 
U_ x  e.  A  ~P { y  |  (
rank `  y )  e.  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897   {cab 2447   ~Pcpw 3962   U_ciun 4291   dom cdm 4852   Oncon0 5441    Fn wfn 5595   ` cfv 5600   R1cr1 8258   rankcrnk 8259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-reg 8132  ax-inf2 8171
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-r1 8260  df-rank 8261
This theorem is referenced by: (None)
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