Theorem rdgisucinc 6656

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

This can be thought of as a generalization of oasuc 6737 and omsuc 6745. (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 6655	. . 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 3386	. . 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 2289	. 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 6653	. . . 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 1278	. . 3 |- ( ph -> ( rec ( F , A ) ` B ) = ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) )
9	8	uneq1d 3382	. 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 6648	. . . . 5 |- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) e. _V )
11	1, 2, 3, 10	syl3anc 1278	. . . 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 5695	. . . . . 6 |- ( x = ( rec ( F , A ) ` B ) -> ( F ` x ) = ( F ` ( rec ( F , A ) ` B ) ) )
15	13, 14	sseq12d 3279	. . . . 5 |- ( x = ( rec ( F , A ) ` B ) -> ( x C_ ( F ` x ) <-> ( rec ( F , A ) ` B ) C_ ( F ` ( rec ( F , A ) ` B ) ) ) )
16	15	spcgv 2912	. . . 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 3399	. . 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 2277	1 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) )

This proof depends on syntax axioms:   -> wi 4   A.wal 1400   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   C_ wss 3220   U_ciun 4012   Oncon0 4508   suc csuc 4510   Fn wfn 5372   ` cfv 5377   reccrdg 6640

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-recs 6576  df-irdg 6641

This theorem is used by:  frecrdg  6679