Theorem rdg0g 6553

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 6537	. . . 4 |- ( x = A -> rec ( F , x ) = rec ( F , A ) )
2	1	fveq1d 5641	. . 3 |- ( x = A -> ( rec ( F , x ) ` (/) ) = ( rec ( F , A ) ` (/) ) )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2246	. 2 |- ( x = A -> ( ( rec ( F , x ) ` (/) ) = x <-> ( rec ( F , A ) ` (/) ) = A ) )
5		vex 2805	. . 3 |- x e. _V
6	5	rdg0 6552	. 2 |- ( rec ( F , x ) ` (/) ) = x
7	4, 6	vtoclg 2864	1 |- ( A e. C -> ( rec ( F , A ) ` (/) ) = A )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   (/)c0 3494   ` cfv 5326   reccrdg 6534

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-recs 6470  df-irdg 6535

This theorem is referenced by:  frecrdg  6573  oa0  6624  oei0  6626