Theorem rdgisuc1 6131

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 6132. (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 4308	. . . 4 |- ( B e. On -> suc B e. On )
5	3, 4	syl 14	. . 3 |- ( ph -> suc B e. On )
6		rdgival 6129	. . 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 1174	. 2 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( A u. U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) ) )
8		df-suc 4189	. . . . . . 7 |- suc B = ( B u. { B } )
9		iuneq1 3738	. . . . . . 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 7	. . . . . 6 |- U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) = U_ x e. ( B u. { B } ) ( F ` ( rec ( F , A ) ` x ) )
11		iunxun 3804	. . . . . 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 2108	. . . . 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 5289	. . . . . . . 8 |- ( x = B -> ( rec ( F , A ) ` x ) = ( rec ( F , A ) ` B ) )
14	13	fveq2d 5293	. . . . . . 7 |- ( x = B -> ( F ` ( rec ( F , A ) ` x ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
15	14	iunxsng 3800	. . . . . 6 |- ( B e. On -> U_ x e. { B } ( F ` ( rec ( F , A ) ` x ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
16	15	uneq2d 3152	. . . . 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 2132	. . . 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 3152	. . 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 2120	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 1289   e. wcel 1438   _Vcvv 2619   u. cun 2995   {csn 3441   U_ciun 3725   Oncon0 4181   suc csuc 4183   Fn wfn 4997   ` cfv 5002   reccrdg 6116

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-recs 6052  df-irdg 6117

This theorem is referenced by:  rdgisucinc  6132