Theorem frecuzrdgrrn 10500

Description: The function R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of S. (Contributed by Jim Kingdon, 28-Mar-2022.)

Hypotheses

Ref	Expression
frec2uz.1	|- ( ph -> C e. ZZ )
frec2uz.2	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
frecuzrdgrrn.a	|- ( ph -> A e. S )
frecuzrdgrrn.f	|- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
frecuzrdgrrn.2	|- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )

Assertion

Ref	Expression
frecuzrdgrrn	|- ( ( ph /\ D e. om ) -> ( R ` D ) e. ( ( ZZ>= ` C ) X. S ) )

Distinct variable groups:   y, A   x, C, y   y, G   x, F, y   x, S, y   ph, x, y

Allowed substitution hints:   A( x)   D( x, y)   R( x, y)   G( x)

Proof of Theorem frecuzrdgrrn

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecuzrdgrrn.2	. . 3 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
2	1	fveq1i 5559	. 2 |- ( R ` D ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` D )
3		frec2uz.1	. . . . . 6 |- ( ph -> C e. ZZ )
4		uzid 9615	. . . . . 6 |- ( C e. ZZ -> C e. ( ZZ>= ` C ) )
5	3, 4	syl 14	. . . . 5 |- ( ph -> C e. ( ZZ>= ` C ) )
6		frecuzrdgrrn.a	. . . . 5 |- ( ph -> A e. S )
7		opelxp 4693	. . . . 5 |- ( <. C , A >. e. ( ( ZZ>= ` C ) X. S ) <-> ( C e. ( ZZ>= ` C ) /\ A e. S ) )
8	5, 6, 7	sylanbrc 417	. . . 4 |- ( ph -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
9	8	adantr 276	. . 3 |- ( ( ph /\ D e. om ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
10		1st2nd2 6233	. . . . . . 7 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
11		fveq2 5558	. . . . . . . 8 |- ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) = ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
12		df-ov 5925	. . . . . . . 8 |- ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) = ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
13	11, 12	eqtr4di 2247	. . . . . . 7 |- ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) = ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) )
14	10, 13	syl 14	. . . . . 6 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) = ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) )
15	14	adantl 277	. . . . 5 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) = ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) )
16		xp1st 6223	. . . . . . 7 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
17	16	adantl 277	. . . . . 6 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
18		xp2nd 6224	. . . . . . 7 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` z ) e. S )
19	18	adantl 277	. . . . . 6 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. S )
20		peano2uz 9657	. . . . . . . 8 |- ( ( 1st ` z ) e. ( ZZ>= ` C ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
21	17, 20	syl 14	. . . . . . 7 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
22		frecuzrdgrrn.f	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
23	22	ralrimivva 2579	. . . . . . . . 9 |- ( ph -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
24	23	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
25		oveq1 5929	. . . . . . . . . . 11 |- ( x = ( 1st ` z ) -> ( x F y ) = ( ( 1st ` z ) F y ) )
26	25	eleq1d 2265	. . . . . . . . . 10 |- ( x = ( 1st ` z ) -> ( ( x F y ) e. S <-> ( ( 1st ` z ) F y ) e. S ) )
27		oveq2 5930	. . . . . . . . . . 11 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) F y ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
28	27	eleq1d 2265	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) F y ) e. S <-> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
29	26, 28	rspc2v 2881	. . . . . . . . 9 |- ( ( ( 1st ` z ) e. ( ZZ>= ` C ) /\ ( 2nd ` z ) e. S ) -> ( A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S -> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
30	17, 19, 29	syl2anc 411	. . . . . . . 8 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S -> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
31	24, 30	mpd 13	. . . . . . 7 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S )
32		opelxp 4693	. . . . . . 7 |- ( <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) <-> ( ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) /\ ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
33	21, 31, 32	sylanbrc 417	. . . . . 6 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
34		oveq1 5929	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( x + 1 ) = ( ( 1st ` z ) + 1 ) )
35	34, 25	opeq12d 3816	. . . . . . 7 |- ( x = ( 1st ` z ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. )
36	27	opeq2d 3815	. . . . . . 7 |- ( y = ( 2nd ` z ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
37		eqid 2196	. . . . . . 7 |- ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) = ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. )
38	35, 36, 37	ovmpog 6057	. . . . . 6 |- ( ( ( 1st ` z ) e. ( ZZ>= ` C ) /\ ( 2nd ` z ) e. S /\ <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
39	17, 19, 33, 38	syl3anc 1249	. . . . 5 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
40	15, 39	eqtrd 2229	. . . 4 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
41	40, 33	eqeltrd 2273	. . 3 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. ( ( ZZ>= ` C ) X. S ) )
42		simpr 110	. . 3 |- ( ( ph /\ D e. om ) -> D e. om )
43	9, 41, 42	freccl 6461	. 2 |- ( ( ph /\ D e. om ) -> (frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` D ) e. ( ( ZZ>= ` C ) X. S ) )
44	2, 43	eqeltrid 2283	1 |- ( ( ph /\ D e. om ) -> ( R ` D ) e. ( ( ZZ>= ` C ) X. S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   <.cop 3625   |-> cmpt 4094   omcom 4626   X. cxp 4661   ` cfv 5258  (class class class)co 5922   e. cmpo 5924   1stc1st 6196   2ndc2nd 6197  freccfrec 6448   1c1 7880   + caddc 7882   ZZcz 9326   ZZ>=cuz 9601

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602

This theorem is referenced by:  frec2uzrdg  10501  frecuzrdgtcl  10504  frecuzrdgsuc  10506