Theorem rdgisucinc 6521

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

This can be thought of as a generalization of oasuc 6600 and omsuc 6608. (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 6520	. . 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 3361	. . 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 2280	. 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 6518	. . . 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 1271	. . 3 |- ( ph -> ( rec ( F , A ) ` B ) = ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) )
9	8	uneq1d 3357	. 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 6513	. . . . 5 |- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) e. _V )
11	1, 2, 3, 10	syl3anc 1271	. . . 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 5623	. . . . . 6 |- ( x = ( rec ( F , A ) ` B ) -> ( F ` x ) = ( F ` ( rec ( F , A ) ` B ) ) )
15	13, 14	sseq12d 3255	. . . . 5 |- ( x = ( rec ( F , A ) ` B ) -> ( x C_ ( F ` x ) <-> ( rec ( F , A ) ` B ) C_ ( F ` ( rec ( F , A ) ` B ) ) ) )
16	15	spcgv 2890	. . . 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 3374	. . 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 2268	1 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) )

Syntax hints:   -> wi 4   A.wal 1393   = wceq 1395   e. wcel 2200   _Vcvv 2799   u. cun 3195   C_ wss 3197   U_ciun 3964   Oncon0 4451   suc csuc 4453   Fn wfn 5309   ` cfv 5314   reccrdg 6505

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-suc 4459  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-recs 6441  df-irdg 6506

This theorem is referenced by:  frecrdg  6544