Theorem rdg0 5997

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 3905	. . . . 5 |- (/) e. _V
2		dmeq 4553	. . . . . . . 8 |- ( g = (/) -> dom g = dom (/) )
3		fveq1 5197	. . . . . . . . 9 |- ( g = (/) -> ( g ` x ) = ( (/) ` x ) )
4	3	fveq2d 5202	. . . . . . . 8 |- ( g = (/) -> ( F ` ( g ` x ) ) = ( F ` ( (/) ` x ) ) )
5	2, 4	iuneq12d 3702	. . . . . . 7 |- ( g = (/) -> U_ x e. dom g ( F ` ( g ` x ) ) = U_ x e. dom (/) ( F ` ( (/) ` x ) ) )
6	5	uneq2d 3126	. . . . . 6 |- ( g = (/) -> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) = ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) )
7		eqid 2081	. . . . . 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 4567	. . . . . . . . . 10 |- dom (/) = (/)
10		iuneq1 3691	. . . . . . . . . 10 |- ( dom (/) = (/) -> U_ x e. dom (/) ( F ` ( (/) ` x ) ) = U_ x e. (/) ( F ` ( (/) ` x ) ) )
11	9, 10	ax-mp 7	. . . . . . . . 9 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) = U_ x e. (/) ( F ` ( (/) ` x ) )
12		0iun 3735	. . . . . . . . 9 |- U_ x e. (/) ( F ` ( (/) ` x ) ) = (/)
13	11, 12	eqtri 2101	. . . . . . . 8 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) = (/)
14	13, 1	eqeltri 2151	. . . . . . 7 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) e. _V
15	8, 14	unex 4194	. . . . . 6 |- ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) e. _V
16	6, 7, 15	fvmpt 5270	. . . . 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 7	. . . 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 2151	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) e. _V
19		df-irdg 5980	. . . 4 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
20	19	tfr0 5960	. . 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 7	. 2 |- ( rec ( F , A ) ` (/) ) = ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) )
22	13	uneq2i 3123	. . . 4 |- ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) = ( A u. (/) )
23	17, 22	eqtri 2101	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = ( A u. (/) )
24		un0 3278	. . 3 |- ( A u. (/) ) = A
25	23, 24	eqtri 2101	. 2 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = A
26	21, 25	eqtri 2101	1 |- ( rec ( F , A ) ` (/) ) = A

Syntax hints:   = wceq 1284   e. wcel 1433   _Vcvv 2601   u. cun 2971   (/)c0 3251   U_ciun 3678   |-> cmpt 3839   dom cdm 4363   ` cfv 4922   reccrdg 5979

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-recs 5943  df-irdg 5980

This theorem is referenced by:  rdg0g  5998  om0  6061