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Theorem findreccl 25918
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 6622 . . 3  |-  ( A  e.  P  ->  ( rec ( G ,  A
) `  (/) )  =  A )
2 eleq1a 2457 . . 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 4792 . . . 4  |-  ( y  e.  om  ->  y  e.  On )
5 fveq2 5669 . . . . . . 7  |-  ( z  =  ( rec ( G ,  A ) `  y )  ->  ( G `  z )  =  ( G `  ( rec ( G ,  A ) `  y
) ) )
65eleq1d 2454 . . . . . 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 2961 . . . . 5  |-  ( ( rec ( G ,  A ) `  y
)  e.  P  -> 
( G `  ( rec ( G ,  A
) `  y )
)  e.  P )
9 rdgsuc 6619 . . . . . 6  |-  ( y  e.  On  ->  ( rec ( G ,  A
) `  suc  y )  =  ( G `  ( rec ( G ,  A ) `  y
) ) )
109eleq1d 2454 . . . . 5  |-  ( y  e.  On  ->  (
( rec ( G ,  A ) `  suc  y )  e.  P  <->  ( G `  ( rec ( G ,  A
) `  y )
)  e.  P ) )
118, 10syl5ibr 213 . . . 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 23 . 2  |-  ( y  e.  om  ->  ( A  e.  P  ->  ( ( rec ( G ,  A ) `  y )  e.  P  ->  ( rec ( G ,  A ) `  suc  y )  e.  P
) ) )
143, 13findfvcl 25917 1  |-  ( C  e.  om  ->  ( A  e.  P  ->  ( rec ( G ,  A ) `  C
)  e.  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   (/)c0 3572   Oncon0 4523   suc csuc 4525   omcom 4786   ` cfv 5395   reccrdg 6604
This theorem is referenced by:  findabrcl  25919
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-recs 6570  df-rdg 6605
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