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Theorem rdg0 7079
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 7075 . . . 4  |-  Lim  dom  rec ( F ,  A
)
2 limomss 6678 . . . 4  |-  ( Lim 
dom  rec ( F ,  A )  ->  om  C_  dom  rec ( F ,  A
) )
31, 2ax-mp 5 . . 3  |-  om  C_  dom  rec ( F ,  A
)
4 peano1 6692 . . 3  |-  (/)  e.  om
53, 4sselii 3486 . 2  |-  (/)  e.  dom  rec ( F ,  A
)
6 eqid 2454 . . 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 7077 . . 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 7064 . 2  |-  ( (/)  e.  dom  rec ( F ,  A )  -> 
( rec ( F ,  A ) `  (/) )  =  A )
105, 9ax-mp 5 1  |-  ( rec ( F ,  A
) `  (/) )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ifcif 3929   U.cuni 4235    |-> cmpt 4497   Lim wlim 4868   dom cdm 4988   ran crn 4989   ` cfv 5570   omcom 6673   reccrdg 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068
This theorem is referenced by:  rdg0g  7085  seqomlem1  7107  seqomlem3  7109  om0  7159  oe0  7164  oev2  7165  r10  8177  aleph0  8438  ackbij2lem2  8611  ackbij2lem3  8612  rdgprc  29467
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