Theorem rdgisucinc 6386

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

This can be thought of as a generalization of oasuc 6465 and omsuc 6473. (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 6385	. . 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 3293	. . 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	eqtr4di 2228	. 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 6383	. . . 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 1238	. . 3 |- ( ph -> ( rec ( F , A ) ` B ) = ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) )
9	8	uneq1d 3289	. 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 6378	. . . . 5 |- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) e. _V )
11	1, 2, 3, 10	syl3anc 1238	. . . 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 5516	. . . . . 6 |- ( x = ( rec ( F , A ) ` B ) -> ( F ` x ) = ( F ` ( rec ( F , A ) ` B ) ) )
15	13, 14	sseq12d 3187	. . . . 5 |- ( x = ( rec ( F , A ) ` B ) -> ( x C_ ( F ` x ) <-> ( rec ( F , A ) ` B ) C_ ( F ` ( rec ( F , A ) ` B ) ) ) )
16	15	spcgv 2825	. . . 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 3306	. . 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 122	. 2 |- ( ph -> ( ( rec ( F , A ) ` B ) u. ( F ` ( rec ( F , A ) ` B ) ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
20	6, 9, 19	3eqtr2d 2216	1 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) )

Syntax hints:   -> wi 4   A.wal 1351   = wceq 1353   e. wcel 2148   _Vcvv 2738   u. cun 3128   C_ wss 3130   U_ciun 3887   Oncon0 4364   suc csuc 4366   Fn wfn 5212   ` cfv 5217   reccrdg 6370

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-recs 6306  df-irdg 6371

This theorem is referenced by:  frecrdg  6409