Theorem frecuzrdgrrn 10630

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 5628	. 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 9736	. . . . . 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 4749	. . . . 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 6321	. . . . . . 7 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
11		fveq2 5627	. . . . . . . 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 6004	. . . . . . . 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 2280	. . . . . . 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 6311	. . . . . . 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 6312	. . . . . . 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 9778	. . . . . . . 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 2612	. . . . . . . . 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 6008	. . . . . . . . . . 11 |- ( x = ( 1st ` z ) -> ( x F y ) = ( ( 1st ` z ) F y ) )
26	25	eleq1d 2298	. . . . . . . . . 10 |- ( x = ( 1st ` z ) -> ( ( x F y ) e. S <-> ( ( 1st ` z ) F y ) e. S ) )
27		oveq2 6009	. . . . . . . . . . 11 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) F y ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
28	27	eleq1d 2298	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) F y ) e. S <-> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
29	26, 28	rspc2v 2920	. . . . . . . . 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 4749	. . . . . . 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 6008	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( x + 1 ) = ( ( 1st ` z ) + 1 ) )
35	34, 25	opeq12d 3865	. . . . . . 7 |- ( x = ( 1st ` z ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. )
36	27	opeq2d 3864	. . . . . . 7 |- ( y = ( 2nd ` z ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
37		eqid 2229	. . . . . . 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 6139	. . . . . 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 1271	. . . . 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 2262	. . . 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 2306	. . 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 6549	. 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 2316	1 |- ( ( ph /\ D e. om ) -> ( R ` D ) e. ( ( ZZ>= ` C ) X. S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   <.cop 3669   |-> cmpt 4145   omcom 4682   X. cxp 4717   ` cfv 5318  (class class class)co 6001   e. cmpo 6003   1stc1st 6284   2ndc2nd 6285  freccfrec 6536   1c1 8000   + caddc 8002   ZZcz 9446   ZZ>=cuz 9722

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  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-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723

This theorem is referenced by:  frec2uzrdg  10631  frecuzrdgtcl  10634  frecuzrdgsuc  10636