Theorem rdgisucinc 6212

Description: Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6290 and omsuc 6298. (Contributed by Jim Kingdon, 29-Aug-2019.)

Hypotheses

Ref	Expression
rdgisuc1.1	|- ( ph -> F Fn _V )
rdgisuc1.2	|- ( ph -> A e. V )
rdgisuc1.3	|- ( ph -> B e. On )
rdgisucinc.inc	|- ( ph -> A. x x C_ ( F ` x ) )

Assertion

Ref	Expression
rdgisucinc	|- ( ph -> ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) )

Distinct variable groups:   x, F   x, A   x, B   x, V

Allowed substitution hint:   ph( x)

Proof of Theorem rdgisucinc

Step	Hyp	Ref	Expression
1		rdgisuc1.1	. . . 4 |- ( ph -> F Fn _V )
2		rdgisuc1.2	. . . 4 |- ( ph -> A e. V )
3		rdgisuc1.3	. . . 4 |- ( ph -> B e. On )
4	1, 2, 3	rdgisuc1 6211	. . 3 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( A u. ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) ) )
5		unass 3180	. . 3 |- ( ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) = ( A u. ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) )
6	4, 5	syl6eqr 2150	. 2 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) )
7		rdgival 6209	. . . 4 |- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) = ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) )
8	1, 2, 3, 7	syl3anc 1184	. . 3 |- ( ph -> ( rec ( F , A ) ` B ) = ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) )
9	8	uneq1d 3176	. 2 |- ( ph -> ( ( rec ( F , A ) ` B ) u. ( F ` ( rec ( F , A ) ` B ) ) ) = ( ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) )
10		rdgexggg 6204	. . . . 5 |- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) e. _V )
11	1, 2, 3, 10	syl3anc 1184	. . . 4 |- ( ph -> ( rec ( F , A ) ` B ) e. _V )
12		rdgisucinc.inc	. . . 4 |- ( ph -> A. x x C_ ( F ` x ) )
13		id 19	. . . . . 6 |- ( x = ( rec ( F , A ) ` B ) -> x = ( rec ( F , A ) ` B ) )
14		fveq2 5353	. . . . . 6 |- ( x = ( rec ( F , A ) ` B ) -> ( F ` x ) = ( F ` ( rec ( F , A ) ` B ) ) )
15	13, 14	sseq12d 3078	. . . . 5 |- ( x = ( rec ( F , A ) ` B ) -> ( x C_ ( F ` x ) <-> ( rec ( F , A ) ` B ) C_ ( F ` ( rec ( F , A ) ` B ) ) ) )
16	15	spcgv 2728	. . . 4 |- ( ( rec ( F , A ) ` B ) e. _V -> ( A. x x C_ ( F ` x ) -> ( rec ( F , A ) ` B ) C_ ( F ` ( rec ( F , A ) ` B ) ) ) )
17	11, 12, 16	sylc 62	. . 3 |- ( ph -> ( rec ( F , A ) ` B ) C_ ( F ` ( rec ( F , A ) ` B ) ) )
18		ssequn1 3193	. . 3 |- ( ( rec ( F , A ) ` B ) C_ ( F ` ( rec ( F , A ) ` B ) ) <-> ( ( rec ( F , A ) ` B ) u. ( F ` ( rec ( F , A ) ` B ) ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
19	17, 18	sylib 121	. 2 |- ( ph -> ( ( rec ( F , A ) ` B ) u. ( F ` ( rec ( F , A ) ` B ) ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
20	6, 9, 19	3eqtr2d 2138	1 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) )

Syntax hints:   -> wi 4   A.wal 1297   = wceq 1299   e. wcel 1448   _Vcvv 2641   u. cun 3019   C_ wss 3021   U_ciun 3760   Oncon0 4223   suc csuc 4225   Fn wfn 5054   ` cfv 5059   reccrdg 6196

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-recs 6132  df-irdg 6197

This theorem is referenced by:  frecrdg  6235