Theorem rdgisuc1 6289

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 6290. (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 4425	. . . 4 |- ( B e. On -> suc B e. On )
5	3, 4	syl 14	. . 3 |- ( ph -> suc B e. On )
6		rdgival 6287	. . 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 1217	. 2 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( A u. U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) ) )
8		df-suc 4301	. . . . . . 7 |- suc B = ( B u. { B } )
9		iuneq1 3834	. . . . . . 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 3900	. . . . . 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 2161	. . . . 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 5429	. . . . . . . 8 |- ( x = B -> ( rec ( F , A ) ` x ) = ( rec ( F , A ) ` B ) )
14	13	fveq2d 5433	. . . . . . 7 |- ( x = B -> ( F ` ( rec ( F , A ) ` x ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
15	14	iunxsng 3896	. . . . . 6 |- ( B e. On -> U_ x e. { B } ( F ` ( rec ( F , A ) ` x ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
16	15	uneq2d 3235	. . . . 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 2185	. . . 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 3235	. . 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 2173	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 1332   e. wcel 1481   _Vcvv 2689   u. cun 3074   {csn 3532   U_ciun 3821   Oncon0 4293   suc csuc 4295   Fn wfn 5126   ` cfv 5131   reccrdg 6274

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-recs 6210  df-irdg 6275

This theorem is referenced by:  rdgisucinc  6290