Theorem rdg0 6648

Description: The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Hypothesis

Ref	Expression
rdg.1	|- A e. _V

Assertion

Ref	Expression
rdg0	|- ( rec ( F , A ) ` (/) ) = A

Proof of Theorem rdg0

Dummy variables x g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4255	. . . . 5 |- (/) e. _V
2		dmeq 4976	. . . . . . . 8 |- ( g = (/) -> dom g = dom (/) )
3		fveq1 5689	. . . . . . . . 9 |- ( g = (/) -> ( g ` x ) = ( (/) ` x ) )
4	3	fveq2d 5694	. . . . . . . 8 |- ( g = (/) -> ( F ` ( g ` x ) ) = ( F ` ( (/) ` x ) ) )
5	2, 4	iuneq12d 4031	. . . . . . 7 |- ( g = (/) -> U_ x e. dom g ( F ` ( g ` x ) ) = U_ x e. dom (/) ( F ` ( (/) ` x ) ) )
6	5	uneq2d 3383	. . . . . 6 |- ( g = (/) -> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) = ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) )
7		eqid 2238	. . . . . 6 |- ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) = ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) )
8		rdg.1	. . . . . . 7 |- A e. _V
9		dm0 4990	. . . . . . . . . 10 |- dom (/) = (/)
10		iuneq1 4020	. . . . . . . . . 10 |- ( dom (/) = (/) -> U_ x e. dom (/) ( F ` ( (/) ` x ) ) = U_ x e. (/) ( F ` ( (/) ` x ) ) )
11	9, 10	ax-mp 5	. . . . . . . . 9 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) = U_ x e. (/) ( F ` ( (/) ` x ) )
12		0iun 4065	. . . . . . . . 9 |- U_ x e. (/) ( F ` ( (/) ` x ) ) = (/)
13	11, 12	eqtri 2259	. . . . . . . 8 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) = (/)
14	13, 1	eqeltri 2311	. . . . . . 7 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) e. _V
15	8, 14	unex 4582	. . . . . 6 |- ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) e. _V
16	6, 7, 15	fvmpt 5776	. . . . 5 |- ( (/) e. _V -> ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) )
17	1, 16	ax-mp 5	. . . 4 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) )
18	17, 15	eqeltri 2311	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) e. _V
19		df-irdg 6631	. . . 4 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
20	19	tfr0 6584	. . 3 |- ( ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) e. _V -> ( rec ( F , A ) ` (/) ) = ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) )
21	18, 20	ax-mp 5	. 2 |- ( rec ( F , A ) ` (/) ) = ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) )
22	13	uneq2i 3380	. . . 4 |- ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) = ( A u. (/) )
23	17, 22	eqtri 2259	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = ( A u. (/) )
24		un0 3556	. . 3 |- ( A u. (/) ) = A
25	23, 24	eqtri 2259	. 2 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = A
26	21, 25	eqtri 2259	1 |- ( rec ( F , A ) ` (/) ) = A

Syntax hints:   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   (/)c0 3520   U_ciun 4007   |-> cmpt 4187   dom cdm 4769   ` cfv 5372   reccrdg 6630

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-recs 6566  df-irdg 6631

This theorem is referenced by:  rdg0g  6649  om0  6721