Theorem rdgisuc1 6352

Description: One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function F other than F Fn _V. Given that, the resulting expression encompasses both the expected successor term ( F ` ( rec ( F , A ) ` B ) ) but also terms that correspond to the initial value A and to limit ordinals U_ x e. B ( F ` ( rec ( F , A ) ` x ) ).

If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6353. (Contributed by Jim Kingdon, 9-Jun-2019.)

Hypotheses

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

Assertion

Ref	Expression
rdgisuc1	|- ( ph -> ( rec ( F , A ) ` suc B ) = ( A u. ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) ) )

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

Allowed substitution hint:   ph( x)

Proof of Theorem rdgisuc1

Step	Hyp	Ref	Expression
1		rdgisuc1.1	. . 3 |- ( ph -> F Fn _V )
2		rdgisuc1.2	. . 3 |- ( ph -> A e. V )
3		rdgisuc1.3	. . . 4 |- ( ph -> B e. On )
4		suceloni 4478	. . . 4 |- ( B e. On -> suc B e. On )
5	3, 4	syl 14	. . 3 |- ( ph -> suc B e. On )
6		rdgival 6350	. . 3 |- ( ( F Fn _V /\ A e. V /\ suc B e. On ) -> ( rec ( F , A ) ` suc B ) = ( A u. U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) ) )
7	1, 2, 5, 6	syl3anc 1228	. 2 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( A u. U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) ) )
8		df-suc 4349	. . . . . . 7 |- suc B = ( B u. { B } )
9		iuneq1 3879	. . . . . . 7 |- ( suc B = ( B u. { B } ) -> U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) = U_ x e. ( B u. { B } ) ( F ` ( rec ( F , A ) ` x ) ) )
10	8, 9	ax-mp 5	. . . . . 6 |- U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) = U_ x e. ( B u. { B } ) ( F ` ( rec ( F , A ) ` x ) )
11		iunxun 3945	. . . . . 6 |- U_ x e. ( B u. { B } ) ( F ` ( rec ( F , A ) ` x ) ) = ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. U_ x e. { B } ( F ` ( rec ( F , A ) ` x ) ) )
12	10, 11	eqtri 2186	. . . . 5 |- U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) = ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. U_ x e. { B } ( F ` ( rec ( F , A ) ` x ) ) )
13		fveq2 5486	. . . . . . . 8 |- ( x = B -> ( rec ( F , A ) ` x ) = ( rec ( F , A ) ` B ) )
14	13	fveq2d 5490	. . . . . . 7 |- ( x = B -> ( F ` ( rec ( F , A ) ` x ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
15	14	iunxsng 3941	. . . . . 6 |- ( B e. On -> U_ x e. { B } ( F ` ( rec ( F , A ) ` x ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
16	15	uneq2d 3276	. . . . 5 |- ( B e. On -> ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. U_ x e. { B } ( F ` ( rec ( F , A ) ` x ) ) ) = ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) )
17	12, 16	syl5eq 2211	. . . 4 |- ( B e. On -> U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) = ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) )
18	17	uneq2d 3276	. . 3 |- ( B e. On -> ( A u. U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) ) = ( A u. ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) ) )
19	3, 18	syl 14	. 2 |- ( ph -> ( A u. U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) ) = ( A u. ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) ) )
20	7, 19	eqtrd 2198	1 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( A u. ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   _Vcvv 2726   u. cun 3114   {csn 3576   U_ciun 3866   Oncon0 4341   suc csuc 4343   Fn wfn 5183   ` cfv 5188   reccrdg 6337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-recs 6273  df-irdg 6338

This theorem is referenced by:  rdgisucinc  6353