Theorem rdgisuc1 6545

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 6546. (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		onsuc 4597	. . . 4 |- ( B e. On -> suc B e. On )
5	3, 4	syl 14	. . 3 |- ( ph -> suc B e. On )
6		rdgival 6543	. . 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 1271	. 2 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( A u. U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) ) )
8		df-suc 4466	. . . . . . 7 |- suc B = ( B u. { B } )
9		iuneq1 3981	. . . . . . 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 4048	. . . . . 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 2250	. . . . 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 5635	. . . . . . . 8 |- ( x = B -> ( rec ( F , A ) ` x ) = ( rec ( F , A ) ` B ) )
14	13	fveq2d 5639	. . . . . . 7 |- ( x = B -> ( F ` ( rec ( F , A ) ` x ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
15	14	iunxsng 4044	. . . . . 6 |- ( B e. On -> U_ x e. { B } ( F ` ( rec ( F , A ) ` x ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
16	15	uneq2d 3359	. . . . 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	eqtrid 2274	. . . 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 3359	. . 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 2262	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 1395   e. wcel 2200   _Vcvv 2800   u. cun 3196   {csn 3667   U_ciun 3968   Oncon0 4458   suc csuc 4460   Fn wfn 5319   ` cfv 5324   reccrdg 6530

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 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-recs 6466  df-irdg 6531

This theorem is referenced by:  rdgisucinc  6546