Theorem rdg0 6454

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 4161	. . . . 5 |- (/) e. _V
2		dmeq 4867	. . . . . . . 8 |- ( g = (/) -> dom g = dom (/) )
3		fveq1 5560	. . . . . . . . 9 |- ( g = (/) -> ( g ` x ) = ( (/) ` x ) )
4	3	fveq2d 5565	. . . . . . . 8 |- ( g = (/) -> ( F ` ( g ` x ) ) = ( F ` ( (/) ` x ) ) )
5	2, 4	iuneq12d 3941	. . . . . . 7 |- ( g = (/) -> U_ x e. dom g ( F ` ( g ` x ) ) = U_ x e. dom (/) ( F ` ( (/) ` x ) ) )
6	5	uneq2d 3318	. . . . . 6 |- ( g = (/) -> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) = ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) )
7		eqid 2196	. . . . . 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 4881	. . . . . . . . . 10 |- dom (/) = (/)
10		iuneq1 3930	. . . . . . . . . 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 3975	. . . . . . . . 9 |- U_ x e. (/) ( F ` ( (/) ` x ) ) = (/)
13	11, 12	eqtri 2217	. . . . . . . 8 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) = (/)
14	13, 1	eqeltri 2269	. . . . . . 7 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) e. _V
15	8, 14	unex 4477	. . . . . 6 |- ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) e. _V
16	6, 7, 15	fvmpt 5641	. . . . 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 2269	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) e. _V
19		df-irdg 6437	. . . 4 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
20	19	tfr0 6390	. . 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 3315	. . . 4 |- ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) = ( A u. (/) )
23	17, 22	eqtri 2217	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = ( A u. (/) )
24		un0 3485	. . 3 |- ( A u. (/) ) = A
25	23, 24	eqtri 2217	. 2 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = A
26	21, 25	eqtri 2217	1 |- ( rec ( F , A ) ` (/) ) = A

Syntax hints:   = wceq 1364   e. wcel 2167   _Vcvv 2763   u. cun 3155   (/)c0 3451   U_ciun 3917   |-> cmpt 4095   dom cdm 4664   ` cfv 5259   reccrdg 6436

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-recs 6372  df-irdg 6437

This theorem is referenced by:  rdg0g  6455  om0  6525