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Theorem findreccl 30146
Description: Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)
Hypothesis
Ref Expression
findreccl.1  |-  ( z  e.  P  ->  ( G `  z )  e.  P )
Assertion
Ref Expression
findreccl  |-  ( C  e.  om  ->  ( A  e.  P  ->  ( rec ( G ,  A ) `  C
)  e.  P ) )
Distinct variable groups:    z, G    z, A    z, P
Allowed substitution hint:    C( z)

Proof of Theorem findreccl
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rdg0g 7085 . . 3  |-  ( A  e.  P  ->  ( rec ( G ,  A
) `  (/) )  =  A )
2 eleq1a 2537 . . 3  |-  ( A  e.  P  ->  (
( rec ( G ,  A ) `  (/) )  =  A  -> 
( rec ( G ,  A ) `  (/) )  e.  P ) )
31, 2mpd 15 . 2  |-  ( A  e.  P  ->  ( rec ( G ,  A
) `  (/) )  e.  P )
4 nnon 6679 . . . 4  |-  ( y  e.  om  ->  y  e.  On )
5 fveq2 5848 . . . . . . 7  |-  ( z  =  ( rec ( G ,  A ) `  y )  ->  ( G `  z )  =  ( G `  ( rec ( G ,  A ) `  y
) ) )
65eleq1d 2523 . . . . . 6  |-  ( z  =  ( rec ( G ,  A ) `  y )  ->  (
( G `  z
)  e.  P  <->  ( G `  ( rec ( G ,  A ) `  y ) )  e.  P ) )
7 findreccl.1 . . . . . 6  |-  ( z  e.  P  ->  ( G `  z )  e.  P )
86, 7vtoclga 3170 . . . . 5  |-  ( ( rec ( G ,  A ) `  y
)  e.  P  -> 
( G `  ( rec ( G ,  A
) `  y )
)  e.  P )
9 rdgsuc 7082 . . . . . 6  |-  ( y  e.  On  ->  ( rec ( G ,  A
) `  suc  y )  =  ( G `  ( rec ( G ,  A ) `  y
) ) )
109eleq1d 2523 . . . . 5  |-  ( y  e.  On  ->  (
( rec ( G ,  A ) `  suc  y )  e.  P  <->  ( G `  ( rec ( G ,  A
) `  y )
)  e.  P ) )
118, 10syl5ibr 221 . . . 4  |-  ( y  e.  On  ->  (
( rec ( G ,  A ) `  y )  e.  P  ->  ( rec ( G ,  A ) `  suc  y )  e.  P
) )
124, 11syl 16 . . 3  |-  ( y  e.  om  ->  (
( rec ( G ,  A ) `  y )  e.  P  ->  ( rec ( G ,  A ) `  suc  y )  e.  P
) )
1312a1d 25 . 2  |-  ( y  e.  om  ->  ( A  e.  P  ->  ( ( rec ( G ,  A ) `  y )  e.  P  ->  ( rec ( G ,  A ) `  suc  y )  e.  P
) ) )
143, 13findfvcl 30145 1  |-  ( C  e.  om  ->  ( A  e.  P  ->  ( rec ( G ,  A ) `  C
)  e.  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   (/)c0 3783   Oncon0 4867   suc csuc 4869   ` 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-rep 4550  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:  findabrcl  30147
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