Theorem rdgisucinc 5995

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

This can be thought of as a generalization of oasuc 6067 and omsuc 6074. (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 5994	. . 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 3129	. . 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 2131	. 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 5992	. . . 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 1169	. . 3 |- ( ph -> ( rec ( F , A ) ` B ) = ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) )
9	8	uneq1d 3125	. 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 5987	. . . . 5 |- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) e. _V )
11	1, 2, 3, 10	syl3anc 1169	. . . 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 5198	. . . . . 6 |- ( x = ( rec ( F , A ) ` B ) -> ( F ` x ) = ( F ` ( rec ( F , A ) ` B ) ) )
15	13, 14	sseq12d 3028	. . . . 5 |- ( x = ( rec ( F , A ) ` B ) -> ( x C_ ( F ` x ) <-> ( rec ( F , A ) ` B ) C_ ( F ` ( rec ( F , A ) ` B ) ) ) )
16	15	spcgv 2685	. . . 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 61	. . 3 |- ( ph -> ( rec ( F , A ) ` B ) C_ ( F ` ( rec ( F , A ) ` B ) ) )
18		ssequn1 3142	. . 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 120	. 2 |- ( ph -> ( ( rec ( F , A ) ` B ) u. ( F ` ( rec ( F , A ) ` B ) ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
20	6, 9, 19	3eqtr2d 2119	1 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) )

Syntax hints:   -> wi 4   A.wal 1282   = wceq 1284   e. wcel 1433   _Vcvv 2601   u. cun 2971   C_ wss 2973   U_ciun 3678   Oncon0 4118   suc csuc 4120   Fn wfn 4917   ` 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-coll 3893  ax-sep 3896  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-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  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-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943  df-irdg 5980

This theorem is referenced by:  frecrdg  6015