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Theorem r1val3 7600
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 7529 . . . . 5  |-  R1  Fn  On
2 fndm 5425 . . . . 5  |-  ( R1  Fn  On  ->  dom  R1  =  On )
31, 2ax-mp 8 . . . 4  |-  dom  R1  =  On
43eleq2i 2422 . . 3  |-  ( A  e.  dom  R1  <->  A  e.  On )
5 r1val1 7548 . . 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 4499 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  A )  ->  x  e.  On )
8 r1val2 7599 . . . . 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 3706 . . 3  |-  ( ( A  e.  On  /\  x  e.  A )  ->  ~P ( R1 `  x )  =  ~P { y  |  (
rank `  y )  e.  x } )
1110iuneq2dv 4007 . 2  |-  ( A  e.  On  ->  U_ x  e.  A  ~P ( R1 `  x )  = 
U_ x  e.  A  ~P { y  |  (
rank `  y )  e.  x } )
126, 11eqtrd 2390 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 1642    e. wcel 1710   {cab 2344   ~Pcpw 3701   U_ciun 3986   Oncon0 4474   dom cdm 4771    Fn wfn 5332   ` cfv 5337   R1cr1 7524   rankcrnk 7525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-reg 7396  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-recs 6475  df-rdg 6510  df-r1 7526  df-rank 7527
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