Theorem frecuzrdgrclt 10307

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. Similar to frecuzrdgrcl 10302 except that S and T need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)

Hypotheses

Ref	Expression
frecuzrdgrclt.c	|- ( ph -> C e. ZZ )
frecuzrdgrclt.a	|- ( ph -> A e. S )
frecuzrdgrclt.t	|- ( ph -> S C_ T )
frecuzrdgrclt.f	|- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
frecuzrdgrclt.r	|- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )

Assertion

Ref	Expression
frecuzrdgrclt	|- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )

Distinct variable groups:   x, C, y   x, F, y   x, S, y   x, T, y   ph, x, y

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

Proof of Theorem frecuzrdgrclt

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1st2nd2 6120	. . . . . . 7 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
2	1	adantl 275	. . . . . 6 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
3	2	fveq2d 5471	. . . . 5 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
4		df-ov 5824	. . . . . . 7 |- ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
5		xp1st 6110	. . . . . . . . 9 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
6	5	adantl 275	. . . . . . . 8 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
7		frecuzrdgrclt.t	. . . . . . . . . 10 |- ( ph -> S C_ T )
8	7	sseld 3127	. . . . . . . . 9 |- ( ph -> ( ( 2nd ` z ) e. S -> ( 2nd ` z ) e. T ) )
9		xp2nd 6111	. . . . . . . . 9 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` z ) e. S )
10	8, 9	impel 278	. . . . . . . 8 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. T )
11		peano2uz 9488	. . . . . . . . . 10 |- ( ( 1st ` z ) e. ( ZZ>= ` C ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
12	6, 11	syl 14	. . . . . . . . 9 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
13		frecuzrdgrclt.f	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
14	13	ralrimivva 2539	. . . . . . . . . . 11 |- ( ph -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
15	14	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
16	9	adantl 275	. . . . . . . . . . 11 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. S )
17		oveq1 5828	. . . . . . . . . . . . 13 |- ( x = ( 1st ` z ) -> ( x F y ) = ( ( 1st ` z ) F y ) )
18	17	eleq1d 2226	. . . . . . . . . . . 12 |- ( x = ( 1st ` z ) -> ( ( x F y ) e. S <-> ( ( 1st ` z ) F y ) e. S ) )
19		oveq2 5829	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) F y ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
20	19	eleq1d 2226	. . . . . . . . . . . 12 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) F y ) e. S <-> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
21	18, 20	rspc2v 2829	. . . . . . . . . . 11 |- ( ( ( 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 ) )
22	6, 16, 21	syl2anc 409	. . . . . . . . . 10 |- ( ( ph /\ 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 ) )
23	15, 22	mpd 13	. . . . . . . . 9 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S )
24		opelxp 4615	. . . . . . . . 9 |- ( <. ( ( 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 ) )
25	12, 23, 24	sylanbrc 414	. . . . . . . 8 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
26		oveq1 5828	. . . . . . . . . 10 |- ( x = ( 1st ` z ) -> ( x + 1 ) = ( ( 1st ` z ) + 1 ) )
27	26, 17	opeq12d 3749	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. )
28	19	opeq2d 3748	. . . . . . . . 9 |- ( y = ( 2nd ` z ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
29		eqid 2157	. . . . . . . . 9 |- ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) = ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. )
30	27, 28, 29	ovmpog 5952	. . . . . . . 8 |- ( ( ( 1st ` z ) e. ( ZZ>= ` C ) /\ ( 2nd ` z ) e. T /\ <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
31	6, 10, 25, 30	syl3anc 1220	. . . . . . 7 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
32	4, 31	eqtr3id 2204	. . . . . 6 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
33	32, 25	eqeltrd 2234	. . . . 5 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) e. ( ( ZZ>= ` C ) X. S ) )
34	3, 33	eqeltrd 2234	. . . 4 |- ( ( ph /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. ( ( ZZ>= ` C ) X. S ) )
35	34	ralrimiva 2530	. . 3 |- ( ph -> A. z e. ( ( ZZ>= ` C ) X. S ) ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. ( ( ZZ>= ` C ) X. S ) )
36		frecuzrdgrclt.c	. . . . 5 |- ( ph -> C e. ZZ )
37		uzid 9447	. . . . 5 |- ( C e. ZZ -> C e. ( ZZ>= ` C ) )
38	36, 37	syl 14	. . . 4 |- ( ph -> C e. ( ZZ>= ` C ) )
39		frecuzrdgrclt.a	. . . 4 |- ( ph -> A e. S )
40		opelxp 4615	. . . 4 |- ( <. C , A >. e. ( ( ZZ>= ` C ) X. S ) <-> ( C e. ( ZZ>= ` C ) /\ A e. S ) )
41	38, 39, 40	sylanbrc 414	. . 3 |- ( ph -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
42		frecfcl 6349	. . 3 |- ( ( A. z e. ( ( ZZ>= ` C ) X. S ) ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. ( ( ZZ>= ` C ) X. S ) /\ <. C , A >. e. ( ( ZZ>= ` C ) X. S ) ) -> frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) : om --> ( ( ZZ>= ` C ) X. S ) )
43	35, 41, 42	syl2anc 409	. 2 |- ( ph -> frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) : om --> ( ( ZZ>= ` C ) X. S ) )
44		frecuzrdgrclt.r	. . 3 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
45	44	feq1i 5311	. 2 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) <-> frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) : om --> ( ( ZZ>= ` C ) X. S ) )
46	43, 45	sylibr 133	1 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   A.wral 2435   C_ wss 3102   <.cop 3563   omcom 4548   X. cxp 4583   -->wf 5165   ` cfv 5169  (class class class)co 5821   e. cmpo 5823   1stc1st 6083   2ndc2nd 6084  freccfrec 6334   1c1 7727   + caddc 7729   ZZcz 9161   ZZ>=cuz 9433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-addcom 7826  ax-addass 7828  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-0id 7834  ax-rnegex 7835  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-ltadd 7842

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-frec 6335  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-inn 8828  df-n0 9085  df-z 9162  df-uz 9434

This theorem is referenced by:  frecuzrdgg  10308  frecuzrdgdomlem  10309  frecuzrdgfunlem  10311  frecuzrdgtclt  10313  frecuzrdg0t  10314  frecuzrdgsuctlem  10315  seq3val  10350  seqvalcd  10351