Theorem rdg0 6631

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 4242	. . . . 5 |- (/) e. _V
2		dmeq 4961	. . . . . . . 8 |- ( g = (/) -> dom g = dom (/) )
3		fveq1 5674	. . . . . . . . 9 |- ( g = (/) -> ( g ` x ) = ( (/) ` x ) )
4	3	fveq2d 5679	. . . . . . . 8 |- ( g = (/) -> ( F ` ( g ` x ) ) = ( F ` ( (/) ` x ) ) )
5	2, 4	iuneq12d 4020	. . . . . . 7 |- ( g = (/) -> U_ x e. dom g ( F ` ( g ` x ) ) = U_ x e. dom (/) ( F ` ( (/) ` x ) ) )
6	5	uneq2d 3377	. . . . . 6 |- ( g = (/) -> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) = ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) )
7		eqid 2234	. . . . . 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 4975	. . . . . . . . . 10 |- dom (/) = (/)
10		iuneq1 4009	. . . . . . . . . 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 4054	. . . . . . . . 9 |- U_ x e. (/) ( F ` ( (/) ` x ) ) = (/)
13	11, 12	eqtri 2255	. . . . . . . 8 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) = (/)
14	13, 1	eqeltri 2307	. . . . . . 7 |- U_ x e. dom (/) ( F ` ( (/) ` x ) ) e. _V
15	8, 14	unex 4567	. . . . . 6 |- ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) e. _V
16	6, 7, 15	fvmpt 5759	. . . . 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 2307	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) e. _V
19		df-irdg 6614	. . . 4 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
20	19	tfr0 6567	. . 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 3374	. . . 4 |- ( A u. U_ x e. dom (/) ( F ` ( (/) ` x ) ) ) = ( A u. (/) )
23	17, 22	eqtri 2255	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = ( A u. (/) )
24		un0 3546	. . 3 |- ( A u. (/) ) = A
25	23, 24	eqtri 2255	. 2 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` (/) ) = A
26	21, 25	eqtri 2255	1 |- ( rec ( F , A ) ` (/) ) = A

Syntax hints:   = wceq 1398   e. wcel 2205   _Vcvv 2815   u. cun 3212   (/)c0 3512   U_ciun 3996   |-> cmpt 4176   dom cdm 4754   ` cfv 5357   reccrdg 6613

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-recs 6549  df-irdg 6614

This theorem is referenced by:  rdg0g  6632  om0  6704