Theorem rdg0g 6532

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.

Step	Hyp	Ref	Expression
1		rdgeq2 6516	. . . 4 |- ( x = A -> rec ( F , x ) = rec ( F , A ) )
2	1	fveq1d 5628	. . 3 |- ( x = A -> ( rec ( F , x ) ` (/) ) = ( rec ( F , A ) ` (/) ) )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2244	. 2 |- ( x = A -> ( ( rec ( F , x ) ` (/) ) = x <-> ( rec ( F , A ) ` (/) ) = A ) )
5		vex 2802	. . 3 |- x e. _V
6	5	rdg0 6531	. 2 |- ( rec ( F , x ) ` (/) ) = x
7	4, 6	vtoclg 2861	1 |- ( A e. C -> ( rec ( F , A ) ` (/) ) = A )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   (/)c0 3491   ` cfv 5317   reccrdg 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-recs 6449  df-irdg 6514

This theorem is referenced by:  frecrdg  6552  oa0  6601  oei0  6603