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Theorem rdglim2a 7159
Description: The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
rdglim2a  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U_ x  e.  B  ( rec ( F ,  A ) `  x
) )
Distinct variable groups:    x, A    x, B    x, F
Allowed substitution hint:    C( x)

Proof of Theorem rdglim2a
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rdglim2 7158 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `
 x ) } )
2 fvex 5891 . . 3  |-  ( rec ( F ,  A
) `  x )  e.  _V
32dfiun2 4336 . 2  |-  U_ x  e.  B  ( rec ( F ,  A ) `
 x )  = 
U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `
 x ) }
41, 3syl6eqr 2488 1  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U_ x  e.  B  ( rec ( F ,  A ) `  x
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {cab 2414   E.wrex 2783   U.cuni 4222   U_ciun 4302   Lim wlim 5443   ` cfv 5601   reccrdg 7135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-wrecs 7036  df-recs 7098  df-rdg 7136
This theorem is referenced by:  oalim  7242  omlim  7243  oelim  7244  alephlim  8496
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