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Theorem r1lim 7698
Description: Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1lim  |-  ( ( A  e.  B  /\  Lim  A )  ->  ( R1 `  A )  = 
U_ x  e.  A  ( R1 `  x ) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem r1lim
StepHypRef Expression
1 limelon 4644 . . 3  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  On )
2 r1fnon 7693 . . . 4  |-  R1  Fn  On
3 fndm 5544 . . . 4  |-  ( R1  Fn  On  ->  dom  R1  =  On )
42, 3ax-mp 8 . . 3  |-  dom  R1  =  On
51, 4syl6eleqr 2527 . 2  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  dom  R1 )
6 r1limg 7697 . 2  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  =  U_ x  e.  A  ( R1 `  x ) )
75, 6sylancom 649 1  |-  ( ( A  e.  B  /\  Lim  A )  ->  ( R1 `  A )  = 
U_ x  e.  A  ( R1 `  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   U_ciun 4093   Oncon0 4581   Lim wlim 4582   dom cdm 4878    Fn wfn 5449   ` cfv 5454   R1cr1 7688
This theorem is referenced by:  r1sdom  7700  r1om  8124  inar1  8650  inatsk  8653  grur1a  8694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668  df-r1 7690
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