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Theorem findreccl 24892
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 6440 . . 3  |-  ( A  e.  P  ->  ( rec ( G ,  A
) `  (/) )  =  A )
2 eleq1a 2352 . . 3  |-  ( A  e.  P  ->  (
( rec ( G ,  A ) `  (/) )  =  A  -> 
( rec ( G ,  A ) `  (/) )  e.  P ) )
31, 2mpd 14 . 2  |-  ( A  e.  P  ->  ( rec ( G ,  A
) `  (/) )  e.  P )
4 nnon 4662 . . . 4  |-  ( y  e.  om  ->  y  e.  On )
5 fveq2 5525 . . . . . . 7  |-  ( z  =  ( rec ( G ,  A ) `  y )  ->  ( G `  z )  =  ( G `  ( rec ( G ,  A ) `  y
) ) )
65eleq1d 2349 . . . . . 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 2849 . . . . 5  |-  ( ( rec ( G ,  A ) `  y
)  e.  P  -> 
( G `  ( rec ( G ,  A
) `  y )
)  e.  P )
9 rdgsuc 6437 . . . . . 6  |-  ( y  e.  On  ->  ( rec ( G ,  A
) `  suc  y )  =  ( G `  ( rec ( G ,  A ) `  y
) ) )
109eleq1d 2349 . . . . 5  |-  ( y  e.  On  ->  (
( rec ( G ,  A ) `  suc  y )  e.  P  <->  ( G `  ( rec ( G ,  A
) `  y )
)  e.  P ) )
118, 10syl5ibr 212 . . . 4  |-  ( y  e.  On  ->  (
( rec ( G ,  A ) `  y )  e.  P  ->  ( rec ( G ,  A ) `  suc  y )  e.  P
) )
124, 11syl 15 . . 3  |-  ( y  e.  om  ->  (
( rec ( G ,  A ) `  y )  e.  P  ->  ( rec ( G ,  A ) `  suc  y )  e.  P
) )
1312a1d 22 . 2  |-  ( y  e.  om  ->  ( A  e.  P  ->  ( ( rec ( G ,  A ) `  y )  e.  P  ->  ( rec ( G ,  A ) `  suc  y )  e.  P
) ) )
143, 13findfvcl 24891 1  |-  ( C  e.  om  ->  ( A  e.  P  ->  ( rec ( G ,  A ) `  C
)  e.  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   (/)c0 3455   Oncon0 4392   suc csuc 4394   omcom 4656   ` cfv 5255   reccrdg 6422
This theorem is referenced by:  findabrcl  24893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423
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