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Theorem rdg0g 6058
Description: The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
Assertion
Ref Expression
rdg0g  |-  ( A  e.  C  ->  ( rec ( F ,  A
) `  (/) )  =  A )

Proof of Theorem rdg0g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rdgeq2 6042 . . . 4  |-  ( x  =  A  ->  rec ( F ,  x )  =  rec ( F ,  A ) )
21fveq1d 5232 . . 3  |-  ( x  =  A  ->  ( rec ( F ,  x
) `  (/) )  =  ( rec ( F ,  A ) `  (/) ) )
3 id 19 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2097 . 2  |-  ( x  =  A  ->  (
( rec ( F ,  x ) `  (/) )  =  x  <->  ( rec ( F ,  A ) `
 (/) )  =  A ) )
5 vex 2613 . . 3  |-  x  e. 
_V
65rdg0 6057 . 2  |-  ( rec ( F ,  x
) `  (/) )  =  x
74, 6vtoclg 2667 1  |-  ( A  e.  C  ->  ( rec ( F ,  A
) `  (/) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   (/)c0 3267   ` cfv 4952   reccrdg 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960  df-recs 5975  df-irdg 6040
This theorem is referenced by:  frecrdg  6078  oa0  6122  oei0  6124
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