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Theorem rdg0 6617
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 6613 . . . 4  |-  Lim  dom  rec ( F ,  A
)
2 limomss 4792 . . . 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 4806 . . 3  |-  (/)  e.  om
53, 4sselii 3290 . 2  |-  (/)  e.  dom  rec ( F ,  A
)
6 eqid 2389 . . 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 6615 . . 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 6602 . 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 1649    e. wcel 1717   _Vcvv 2901    C_ wss 3265   (/)c0 3573   ifcif 3684   U.cuni 3959    e. cmpt 4209   Lim wlim 4525   omcom 4787   dom cdm 4820   ran crn 4821   ` cfv 5396   reccrdg 6605
This theorem is referenced by:  rdg0g  6623  seqomlem1  6645  seqomlem3  6647  abianfplem  6653  om0  6699  oe0  6704  oev2  6705  r10  7629  aleph0  7882  ackbij2lem2  8055  ackbij2lem3  8056  rdgprc  25177
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-recs 6571  df-rdg 6606
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