Theorem rdg0 6440

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 4156	. . . . 5 |- (/) e. _V
2		dmeq 4862	. . . . . . . 8 |- ( g = (/) -> dom g = dom (/) )
3		fveq1 5553	. . . . . . . . 9 |- ( g = (/) -> ( g ` x ) = ( (/) ` x ) )
4	3	fveq2d 5558	. . . . . . . 8 |- ( g = (/) -> ( F ` ( g ` x ) ) = ( F ` ( (/) ` x ) ) )
5	2, 4	iuneq12d 3936	. . . . . . 7 |- ( g = (/) -> U_ x e. dom g ( F ` ( g ` x ) ) = U_ x e. dom (/) ( F ` ( (/) ` x ) ) )
6	5	uneq2d 3313	. . . . . 6 |- ( g = (/) -> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) = ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) )
7		eqid 2193	. . . . . 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 4876	. . . . . . . . . 10 |- dom (/) = (/)
10		iuneq1 3925	. . . . . . . . . 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 3970	. . . . . . . . 9 |- U_ x e. (/) ( F ` ( (/) ` x ) ) = (/)
13	11, 12	eqtri 2214	. . . . . . . 8 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) = (/)
14	13, 1	eqeltri 2266	. . . . . . 7 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) e. _V
15	8, 14	unex 4472	. . . . . 6 |- ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) e. _V
16	6, 7, 15	fvmpt 5634	. . . . 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 2266	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) e. _V
19		df-irdg 6423	. . . 4 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
20	19	tfr0 6376	. . 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 3310	. . . 4 |- ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) = ( A u. (/) )
23	17, 22	eqtri 2214	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = ( A u. (/) )
24		un0 3480	. . 3 |- ( A u. (/) ) = A
25	23, 24	eqtri 2214	. 2 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = A
26	21, 25	eqtri 2214	1 |- ( rec ( F , A ) ` (/) ) = A

Syntax hints:   = wceq 1364   e. wcel 2164   _Vcvv 2760   u. cun 3151   (/)c0 3446   U_ciun 3912   |-> cmpt 4090   dom cdm 4659   ` cfv 5254   reccrdg 6422

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-recs 6358  df-irdg 6423

This theorem is referenced by:  rdg0g  6441  om0  6511