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Theorem rdg0 6450
Description: The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
rdg.1  |-  A  e. 
_V
Assertion
Ref Expression
rdg0  |-  ( rec ( F ,  A
) `  (/) )  =  A

Proof of Theorem rdg0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rdgdmlim 6446 . . . 4  |-  Lim  dom  rec ( F ,  A
)
2 limomss 4677 . . . 4  |-  ( Lim 
dom  rec ( F ,  A )  ->  om  C_  dom  rec ( F ,  A
) )
31, 2ax-mp 8 . . 3  |-  om  C_  dom  rec ( F ,  A
)
4 peano1 4691 . . 3  |-  (/)  e.  om
53, 4sselii 3190 . 2  |-  (/)  e.  dom  rec ( F ,  A
)
6 eqid 2296 . . 3  |-  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( F `  ( x `  U. dom  x ) ) ) ) )  =  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( F `  ( x `
 U. dom  x
) ) ) ) )
7 rdgvalg 6448 . . 3  |-  ( y  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  y )  =  ( ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( F `  ( x `
 U. dom  x
) ) ) ) ) `  ( rec ( F ,  A
)  |`  y ) ) )
8 rdg.1 . . 3  |-  A  e. 
_V
96, 7, 8tz7.44-1 6435 . 2  |-  ( (/)  e.  dom  rec ( F ,  A )  -> 
( rec ( F ,  A ) `  (/) )  =  A )
105, 9ax-mp 8 1  |-  ( rec ( F ,  A
) `  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ifcif 3578   U.cuni 3843    e. cmpt 4093   Lim wlim 4409   omcom 4672   dom cdm 4705   ran crn 4706   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  rdg0g  6456  seqomlem1  6478  seqomlem3  6480  abianfplem  6486  om0  6532  oe0  6537  oev2  6538  r10  7456  aleph0  7709  ackbij2lem2  7882  ackbij2lem3  7883  rdgprc  24222  vtare  25988
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-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-om 4673  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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