Theorem rdgisuc1 6549

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 6550. (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 4599	. . . 4 |- ( B e. On -> suc B e. On )
5	3, 4	syl 14	. . 3 |- ( ph -> suc B e. On )
6		rdgival 6547	. . 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 1273	. 2 |- ( ph -> ( rec ( F , A ) ` suc B ) = ( A u. U_ x e. suc B ( F ` ( rec ( F , A ) ` x ) ) ) )
8		df-suc 4468	. . . . . . 7 |- suc B = ( B u. { B } )
9		iuneq1 3983	. . . . . . 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 4050	. . . . . 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 2252	. . . . 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 5639	. . . . . . . 8 |- ( x = B -> ( rec ( F , A ) ` x ) = ( rec ( F , A ) ` B ) )
14	13	fveq2d 5643	. . . . . . 7 |- ( x = B -> ( F ` ( rec ( F , A ) ` x ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
15	14	iunxsng 4046	. . . . . 6 |- ( B e. On -> U_ x e. { B } ( F ` ( rec ( F , A ) ` x ) ) = ( F ` ( rec ( F , A ) ` B ) ) )
16	15	uneq2d 3361	. . . . 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 2276	. . . 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 3361	. . 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 2264	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 1397   e. wcel 2202   _Vcvv 2802   u. cun 3198   {csn 3669   U_ciun 3970   Oncon0 4460   suc csuc 4462   Fn wfn 5321   ` cfv 5326   reccrdg 6534

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-recs 6470  df-irdg 6535

This theorem is referenced by:  rdgisucinc  6550