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Theorem findreccl 24964
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 6456 . . 3  |-  ( A  e.  P  ->  ( rec ( G ,  A
) `  (/) )  =  A )
2 eleq1a 2365 . . 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 4678 . . . 4  |-  ( y  e.  om  ->  y  e.  On )
5 fveq2 5541 . . . . . . 7  |-  ( z  =  ( rec ( G ,  A ) `  y )  ->  ( G `  z )  =  ( G `  ( rec ( G ,  A ) `  y
) ) )
65eleq1d 2362 . . . . . 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 2862 . . . . 5  |-  ( ( rec ( G ,  A ) `  y
)  e.  P  -> 
( G `  ( rec ( G ,  A
) `  y )
)  e.  P )
9 rdgsuc 6453 . . . . . 6  |-  ( y  e.  On  ->  ( rec ( G ,  A
) `  suc  y )  =  ( G `  ( rec ( G ,  A ) `  y
) ) )
109eleq1d 2362 . . . . 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 24963 1  |-  ( C  e.  om  ->  ( A  e.  P  ->  ( rec ( G ,  A ) `  C
)  e.  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   (/)c0 3468   Oncon0 4408   suc csuc 4410   omcom 4672   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  findabrcl  24965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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