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Theorem rdglim2a 6462
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 6461 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `
 x ) } )
2 fvex 5555 . . 3  |-  ( rec ( F ,  A
) `  x )  e.  _V
32dfiun2 3953 . 2  |-  U_ x  e.  B  ( rec ( F ,  A ) `
 x )  = 
U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `
 x ) }
41, 3syl6eqr 2346 1  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U_ x  e.  B  ( rec ( F ,  A ) `  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   U.cuni 3843   U_ciun 3921   Lim wlim 4409   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  abianfplem  6486  oalim  6547  omlim  6548  oelim  6549  alephlim  7710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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