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Theorem rdg0g 6456
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 6441 . . . 4  |-  ( x  =  A  ->  rec ( F ,  x )  =  rec ( F ,  A ) )
21fveq1d 5543 . . 3  |-  ( x  =  A  ->  ( rec ( F ,  x
) `  (/) )  =  ( rec ( F ,  A ) `  (/) ) )
3 id 19 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2310 . 2  |-  ( x  =  A  ->  (
( rec ( F ,  x ) `  (/) )  =  x  <->  ( rec ( F ,  A ) `
 (/) )  =  A ) )
5 vex 2804 . . 3  |-  x  e. 
_V
65rdg0 6450 . 2  |-  ( rec ( F ,  x
) `  (/) )  =  x
74, 6vtoclg 2856 1  |-  ( A  e.  C  ->  ( rec ( F ,  A
) `  (/) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   (/)c0 3468   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  fr0g  6464  oa0  6531  findreccl  24964
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-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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