Theorem frecuzrdgrrn 9194

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, 27-May-2020.)

Hypotheses

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

Proof of Theorem frecuzrdgrrn

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzrdg.2	. . 3 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
2	1	fveq1i 5179	. 2 |- ( R ` D ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` D )
3		zex 8254	. . . . . . . 8 |- ZZ e. _V
4		uzssz 8492	. . . . . . . 8 |- ( ZZ>= ` C ) C_ ZZ
5	3, 4	ssexi 3895	. . . . . . 7 |- ( ZZ>= ` C ) e. _V
6	5	a1i 9	. . . . . 6 |- ( ( ph /\ D e. om ) -> ( ZZ>= ` C ) e. _V )
7		uzrdg.s	. . . . . . 7 |- ( ph -> S e. V )
8	7	adantr 261	. . . . . 6 |- ( ( ph /\ D e. om ) -> S e. V )
9		mpt2exga 5835	. . . . . 6 |- ( ( ( ZZ>= ` C ) e. _V /\ S e. V ) -> ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) e. _V )
10	6, 8, 9	syl2anc 391	. . . . 5 |- ( ( ph /\ D e. om ) -> ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) e. _V )
11		vex 2560	. . . . . 6 |- z e. _V
12	11	a1i 9	. . . . 5 |- ( ( ph /\ D e. om ) -> z e. _V )
13		fvexg 5194	. . . . 5 |- ( ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) e. _V /\ z e. _V ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. _V )
14	10, 12, 13	syl2anc 391	. . . 4 |- ( ( ph /\ D e. om ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. _V )
15	14	alrimiv 1754	. . 3 |- ( ( ph /\ D e. om ) -> A. z ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. _V )
16		frec2uz.1	. . . . . 6 |- ( ph -> C e. ZZ )
17		uzid 8487	. . . . . 6 |- ( C e. ZZ -> C e. ( ZZ>= ` C ) )
18	16, 17	syl 14	. . . . 5 |- ( ph -> C e. ( ZZ>= ` C ) )
19		uzrdg.a	. . . . 5 |- ( ph -> A e. S )
20		opelxp 4374	. . . . 5 |- ( <. C , A >. e. ( ( ZZ>= ` C ) X. S ) <-> ( C e. ( ZZ>= ` C ) /\ A e. S ) )
21	18, 19, 20	sylanbrc 394	. . . 4 |- ( ph -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
22	21	adantr 261	. . 3 |- ( ( ph /\ D e. om ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
23		1st2nd2 5801	. . . . . . 7 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
24		fveq2 5178	. . . . . . . 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 ) >. ) )
25		df-ov 5515	. . . . . . . 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 ) >. )
26	24, 25	syl6eqr 2090	. . . . . . 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 ) ) )
27	23, 26	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 ) ) )
28	27	adantl 262	. . . . 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 ) ) )
29		xp1st 5792	. . . . . . 7 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
30	29	adantl 262	. . . . . 6 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
31		xp2nd 5793	. . . . . . 7 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` z ) e. S )
32	31	adantl 262	. . . . . 6 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. S )
33		peano2uz 8526	. . . . . . . 8 |- ( ( 1st ` z ) e. ( ZZ>= ` C ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
34	30, 33	syl 14	. . . . . . 7 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
35		uzrdg.f	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
36	35	ralrimivva 2401	. . . . . . . . 9 |- ( ph -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
37	36	ad2antrr 457	. . . . . . . 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 )
38		oveq1 5519	. . . . . . . . . . 11 |- ( x = ( 1st ` z ) -> ( x F y ) = ( ( 1st ` z ) F y ) )
39	38	eleq1d 2106	. . . . . . . . . 10 |- ( x = ( 1st ` z ) -> ( ( x F y ) e. S <-> ( ( 1st ` z ) F y ) e. S ) )
40		oveq2 5520	. . . . . . . . . . 11 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) F y ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
41	40	eleq1d 2106	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) F y ) e. S <-> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
42	39, 41	rspc2v 2662	. . . . . . . . 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 ) )
43	30, 32, 42	syl2anc 391	. . . . . . . 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 ) )
44	37, 43	mpd 13	. . . . . . 7 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S )
45		opelxp 4374	. . . . . . 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 ) )
46	34, 44, 45	sylanbrc 394	. . . . . 6 |- ( ( ( ph /\ D e. om ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
47		oveq1 5519	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( x + 1 ) = ( ( 1st ` z ) + 1 ) )
48	47, 38	opeq12d 3557	. . . . . . 7 |- ( x = ( 1st ` z ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. )
49	40	opeq2d 3556	. . . . . . 7 |- ( y = ( 2nd ` z ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
50		eqid 2040	. . . . . . 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 ) >. )
51	48, 49, 50	ovmpt2g 5635	. . . . . 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 ) ) >. )
52	30, 32, 46, 51	syl3anc 1135	. . . . 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 ) ) >. )
53	28, 52	eqtrd 2072	. . . 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 ) ) >. )
54	53, 46	eqeltrd 2114	. . 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 ) )
55		simpr 103	. . 3 |- ( ( ph /\ D e. om ) -> D e. om )
56	15, 22, 54, 55	freccl 5993	. 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 ) )
57	2, 56	syl5eqel 2124	1 |- ( ( ph /\ D e. om ) -> ( R ` D ) e. ( ( ZZ>= ` C ) X. S ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   A.wral 2306   _Vcvv 2557   <.cop 3378   |-> cmpt 3818   omcom 4313   X. cxp 4343   ` cfv 4902  (class class class)co 5512   |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766  freccfrec 5977   1c1 6890   + caddc 6892   ZZcz 8245   ZZ>=cuz 8473

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474

This theorem is referenced by:  frec2uzrdg  9195  frecuzrdgfn  9198  frecuzrdgcl  9199  frecuzrdgsuc  9201