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Theorem r1lim 7654
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 4604 . . 3  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  On )
2 r1fnon 7649 . . . 4  |-  R1  Fn  On
3 fndm 5503 . . . 4  |-  ( R1  Fn  On  ->  dom  R1  =  On )
42, 3ax-mp 8 . . 3  |-  dom  R1  =  On
51, 4syl6eleqr 2495 . 2  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  dom  R1 )
6 r1limg 7653 . 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 1649    e. wcel 1721   U_ciun 4053   Oncon0 4541   Lim wlim 4542   dom cdm 4837    Fn wfn 5408   ` cfv 5413   R1cr1 7644
This theorem is referenced by:  r1sdom  7656  r1om  8080  inar1  8606  inatsk  8609  grur1a  8650
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-r1 7646
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