Theorem frecuzrdgg 10351

Description: Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating R at a natural number gives an ordered pair whose first element is the mapping of that natural number via G. (Contributed by Jim Kingdon, 23-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 >. )
frecuzrdgg.n	|- ( ph -> N e. om )
frecuzrdgg.g	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )

Assertion

Ref	Expression
frecuzrdgg	|- ( ph -> ( 1st ` ( R ` N ) ) = ( G ` N ) )

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

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

Proof of Theorem frecuzrdgg

Dummy variables z k w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecuzrdgg.n	. 2 |- ( ph -> N e. om )
2		fveq2 5486	. . . . . 6 |- ( w = (/) -> ( R ` w ) = ( R ` (/) ) )
3	2	fveq2d 5490	. . . . 5 |- ( w = (/) -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` (/) ) ) )
4		fveq2 5486	. . . . 5 |- ( w = (/) -> ( G ` w ) = ( G ` (/) ) )
5	3, 4	eqeq12d 2180	. . . 4 |- ( w = (/) -> ( ( 1st ` ( R ` w ) ) = ( G ` w ) <-> ( 1st ` ( R ` (/) ) ) = ( G ` (/) ) ) )
6	5	imbi2d 229	. . 3 |- ( w = (/) -> ( ( ph -> ( 1st ` ( R ` w ) ) = ( G ` w ) ) <-> ( ph -> ( 1st ` ( R ` (/) ) ) = ( G ` (/) ) ) ) )
7		fveq2 5486	. . . . . 6 |- ( w = k -> ( R ` w ) = ( R ` k ) )
8	7	fveq2d 5490	. . . . 5 |- ( w = k -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` k ) ) )
9		fveq2 5486	. . . . 5 |- ( w = k -> ( G ` w ) = ( G ` k ) )
10	8, 9	eqeq12d 2180	. . . 4 |- ( w = k -> ( ( 1st ` ( R ` w ) ) = ( G ` w ) <-> ( 1st ` ( R ` k ) ) = ( G ` k ) ) )
11	10	imbi2d 229	. . 3 |- ( w = k -> ( ( ph -> ( 1st ` ( R ` w ) ) = ( G ` w ) ) <-> ( ph -> ( 1st ` ( R ` k ) ) = ( G ` k ) ) ) )
12		fveq2 5486	. . . . . 6 |- ( w = suc k -> ( R ` w ) = ( R ` suc k ) )
13	12	fveq2d 5490	. . . . 5 |- ( w = suc k -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` suc k ) ) )
14		fveq2 5486	. . . . 5 |- ( w = suc k -> ( G ` w ) = ( G ` suc k ) )
15	13, 14	eqeq12d 2180	. . . 4 |- ( w = suc k -> ( ( 1st ` ( R ` w ) ) = ( G ` w ) <-> ( 1st ` ( R ` suc k ) ) = ( G ` suc k ) ) )
16	15	imbi2d 229	. . 3 |- ( w = suc k -> ( ( ph -> ( 1st ` ( R ` w ) ) = ( G ` w ) ) <-> ( ph -> ( 1st ` ( R ` suc k ) ) = ( G ` suc k ) ) ) )
17		fveq2 5486	. . . . . 6 |- ( w = N -> ( R ` w ) = ( R ` N ) )
18	17	fveq2d 5490	. . . . 5 |- ( w = N -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` N ) ) )
19		fveq2 5486	. . . . 5 |- ( w = N -> ( G ` w ) = ( G ` N ) )
20	18, 19	eqeq12d 2180	. . . 4 |- ( w = N -> ( ( 1st ` ( R ` w ) ) = ( G ` w ) <-> ( 1st ` ( R ` N ) ) = ( G ` N ) ) )
21	20	imbi2d 229	. . 3 |- ( w = N -> ( ( ph -> ( 1st ` ( R ` w ) ) = ( G ` w ) ) <-> ( ph -> ( 1st ` ( R ` N ) ) = ( G ` N ) ) ) )
22		frecuzrdgrclt.c	. . . . 5 |- ( ph -> C e. ZZ )
23		frecuzrdgrclt.a	. . . . 5 |- ( ph -> A e. S )
24		op1stg 6118	. . . . 5 |- ( ( C e. ZZ /\ A e. S ) -> ( 1st ` <. C , A >. ) = C )
25	22, 23, 24	syl2anc 409	. . . 4 |- ( ph -> ( 1st ` <. C , A >. ) = C )
26		frecuzrdgrclt.r	. . . . . . 7 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
27	26	fveq1i 5487	. . . . . 6 |- ( R ` (/) ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` (/) )
28		opexg 4206	. . . . . . . 8 |- ( ( C e. ZZ /\ A e. S ) -> <. C , A >. e. _V )
29		frec0g 6365	. . . . . . . 8 |- ( <. C , A >. e. _V -> (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` (/) ) = <. C , A >. )
30	28, 29	syl 14	. . . . . . 7 |- ( ( C e. ZZ /\ A e. S ) -> (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` (/) ) = <. C , A >. )
31	22, 23, 30	syl2anc 409	. . . . . 6 |- ( ph -> (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` (/) ) = <. C , A >. )
32	27, 31	syl5eq 2211	. . . . 5 |- ( ph -> ( R ` (/) ) = <. C , A >. )
33	32	fveq2d 5490	. . . 4 |- ( ph -> ( 1st ` ( R ` (/) ) ) = ( 1st ` <. C , A >. ) )
34		frecuzrdgg.g	. . . . 5 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
35	22, 34	frec2uz0d 10334	. . . 4 |- ( ph -> ( G ` (/) ) = C )
36	25, 33, 35	3eqtr4d 2208	. . 3 |- ( ph -> ( 1st ` ( R ` (/) ) ) = ( G ` (/) ) )
37	22, 34	frec2uzf1od 10341	. . . . . . . . . . 11 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
38		f1of 5432	. . . . . . . . . . 11 |- ( G : om -1-1-onto-> ( ZZ>= ` C ) -> G : om --> ( ZZ>= ` C ) )
39	37, 38	syl 14	. . . . . . . . . 10 |- ( ph -> G : om --> ( ZZ>= ` C ) )
40	39	ad2antlr 481	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> G : om --> ( ZZ>= ` C ) )
41		simpll 519	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> k e. om )
42	40, 41	ffvelrnd 5621	. . . . . . . 8 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( G ` k ) e. ( ZZ>= ` C ) )
43		peano2uz 9521	. . . . . . . 8 |- ( ( G ` k ) e. ( ZZ>= ` C ) -> ( ( G ` k ) + 1 ) e. ( ZZ>= ` C ) )
44	42, 43	syl 14	. . . . . . 7 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( ( G ` k ) + 1 ) e. ( ZZ>= ` C ) )
45		oveq2 5850	. . . . . . . . 9 |- ( y = ( 2nd ` ( R ` k ) ) -> ( ( G ` k ) F y ) = ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) )
46	45	eleq1d 2235	. . . . . . . 8 |- ( y = ( 2nd ` ( R ` k ) ) -> ( ( ( G ` k ) F y ) e. S <-> ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) e. S ) )
47		oveq1 5849	. . . . . . . . . . 11 |- ( x = ( G ` k ) -> ( x F y ) = ( ( G ` k ) F y ) )
48	47	eleq1d 2235	. . . . . . . . . 10 |- ( x = ( G ` k ) -> ( ( x F y ) e. S <-> ( ( G ` k ) F y ) e. S ) )
49	48	ralbidv 2466	. . . . . . . . 9 |- ( x = ( G ` k ) -> ( A. y e. S ( x F y ) e. S <-> A. y e. S ( ( G ` k ) F y ) e. S ) )
50		frecuzrdgrclt.f	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
51	50	ralrimivva 2548	. . . . . . . . . 10 |- ( ph -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
52	51	ad2antlr 481	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
53	49, 52, 42	rspcdva 2835	. . . . . . . 8 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> A. y e. S ( ( G ` k ) F y ) e. S )
54		frecuzrdgrclt.t	. . . . . . . . . . . 12 |- ( ph -> S C_ T )
55	22, 23, 54, 50, 26	frecuzrdgrclt 10350	. . . . . . . . . . 11 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
56	55	ad2antlr 481	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> R : om --> ( ( ZZ>= ` C ) X. S ) )
57	56, 41	ffvelrnd 5621	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( R ` k ) e. ( ( ZZ>= ` C ) X. S ) )
58		xp2nd 6134	. . . . . . . . 9 |- ( ( R ` k ) e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` ( R ` k ) ) e. S )
59	57, 58	syl 14	. . . . . . . 8 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 2nd ` ( R ` k ) ) e. S )
60	46, 53, 59	rspcdva 2835	. . . . . . 7 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) e. S )
61		op1stg 6118	. . . . . . 7 |- ( ( ( ( G ` k ) + 1 ) e. ( ZZ>= ` C ) /\ ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) e. S ) -> ( 1st ` <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. ) = ( ( G ` k ) + 1 ) )
62	44, 60, 61	syl2anc 409	. . . . . 6 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 1st ` <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. ) = ( ( G ` k ) + 1 ) )
63		1st2nd2 6143	. . . . . . . . . . . . . . . 16 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
64	63	adantl 275	. . . . . . . . . . . . . . 15 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
65	64	fveq2d 5490	. . . . . . . . . . . . . 14 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ 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 ) >. ) )
66		df-ov 5845	. . . . . . . . . . . . . . . 16 |- ( ( 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 ) >. )
67		xp1st 6133	. . . . . . . . . . . . . . . . . 18 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
68	67	adantl 275	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
69	54	ad3antlr 485	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> S C_ T )
70		xp2nd 6134	. . . . . . . . . . . . . . . . . . 19 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` z ) e. S )
71	70	adantl 275	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. S )
72	69, 71	sseldd 3143	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. T )
73		peano2uz 9521	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1st ` z ) e. ( ZZ>= ` C ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
74	68, 73	syl 14	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
75		oveq2 5850	. . . . . . . . . . . . . . . . . . . 20 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) F y ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
76	75	eleq1d 2235	. . . . . . . . . . . . . . . . . . 19 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) F y ) e. S <-> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
77		oveq1 5849	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x = ( 1st ` z ) -> ( x F y ) = ( ( 1st ` z ) F y ) )
78	77	eleq1d 2235	. . . . . . . . . . . . . . . . . . . . 21 |- ( x = ( 1st ` z ) -> ( ( x F y ) e. S <-> ( ( 1st ` z ) F y ) e. S ) )
79	78	ralbidv 2466	. . . . . . . . . . . . . . . . . . . 20 |- ( x = ( 1st ` z ) -> ( A. y e. S ( x F y ) e. S <-> A. y e. S ( ( 1st ` z ) F y ) e. S ) )
80	51	ad3antlr 485	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
81	79, 80, 68	rspcdva 2835	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> A. y e. S ( ( 1st ` z ) F y ) e. S )
82	76, 81, 71	rspcdva 2835	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S )
83		opelxpi 4636	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) /\ ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
84	74, 82, 83	syl2anc 409	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
85		oveq1 5849	. . . . . . . . . . . . . . . . . . 19 |- ( x = ( 1st ` z ) -> ( x + 1 ) = ( ( 1st ` z ) + 1 ) )
86	85, 77	opeq12d 3766	. . . . . . . . . . . . . . . . . 18 |- ( x = ( 1st ` z ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. )
87	75	opeq2d 3765	. . . . . . . . . . . . . . . . . 18 |- ( y = ( 2nd ` z ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
88		eqid 2165	. . . . . . . . . . . . . . . . . 18 |- ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) = ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. )
89	86, 87, 88	ovmpog 5976	. . . . . . . . . . . . . . . . 17 |- ( ( ( 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 ) ) >. )
90	68, 72, 84, 89	syl3anc 1228	. . . . . . . . . . . . . . . 16 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ 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 ) ) >. )
91	66, 90	eqtr3id 2213	. . . . . . . . . . . . . . 15 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ 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 ) ) >. )
92	91, 84	eqeltrd 2243	. . . . . . . . . . . . . 14 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ 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 ) )
93	65, 92	eqeltrd 2243	. . . . . . . . . . . . 13 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. ( ( ZZ>= ` C ) X. S ) )
94	93	ralrimiva 2539	. . . . . . . . . . . 12 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> 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 ) )
95		uzid 9480	. . . . . . . . . . . . . . 15 |- ( C e. ZZ -> C e. ( ZZ>= ` C ) )
96	22, 95	syl 14	. . . . . . . . . . . . . 14 |- ( ph -> C e. ( ZZ>= ` C ) )
97		opelxpi 4636	. . . . . . . . . . . . . 14 |- ( ( C e. ( ZZ>= ` C ) /\ A e. S ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
98	96, 23, 97	syl2anc 409	. . . . . . . . . . . . 13 |- ( ph -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
99	98	ad2antlr 481	. . . . . . . . . . . 12 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
100		frecsuc 6375	. . . . . . . . . . . 12 |- ( ( 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 ) /\ k e. om ) -> (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc k ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` k ) ) )
101	94, 99, 41, 100	syl3anc 1228	. . . . . . . . . . 11 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc k ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` k ) ) )
102	26	fveq1i 5487	. . . . . . . . . . 11 |- ( R ` suc k ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc k )
103	26	fveq1i 5487	. . . . . . . . . . . 12 |- ( R ` k ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` k )
104	103	fveq2i 5489	. . . . . . . . . . 11 |- ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` k ) ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` k ) )
105	101, 102, 104	3eqtr4g 2224	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( R ` suc k ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` k ) ) )
106		1st2nd2 6143	. . . . . . . . . . . 12 |- ( ( R ` k ) e. ( ( ZZ>= ` C ) X. S ) -> ( R ` k ) = <. ( 1st ` ( R ` k ) ) , ( 2nd ` ( R ` k ) ) >. )
107	57, 106	syl 14	. . . . . . . . . . 11 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( R ` k ) = <. ( 1st ` ( R ` k ) ) , ( 2nd ` ( R ` k ) ) >. )
108	107	fveq2d 5490	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` k ) ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` ( R ` k ) ) , ( 2nd ` ( R ` k ) ) >. ) )
109	105, 108	eqtrd 2198	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( R ` suc k ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` ( R ` k ) ) , ( 2nd ` ( R ` k ) ) >. ) )
110		simpr 109	. . . . . . . . . . 11 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 1st ` ( R ` k ) ) = ( G ` k ) )
111	110	opeq1d 3764	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> <. ( 1st ` ( R ` k ) ) , ( 2nd ` ( R ` k ) ) >. = <. ( G ` k ) , ( 2nd ` ( R ` k ) ) >. )
112	111	fveq2d 5490	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` ( R ` k ) ) , ( 2nd ` ( R ` k ) ) >. ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` k ) , ( 2nd ` ( R ` k ) ) >. ) )
113	109, 112	eqtrd 2198	. . . . . . . 8 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( R ` suc k ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` k ) , ( 2nd ` ( R ` k ) ) >. ) )
114		df-ov 5845	. . . . . . . . 9 |- ( ( G ` k ) ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` k ) ) ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` k ) , ( 2nd ` ( R ` k ) ) >. )
115	54	ad2antlr 481	. . . . . . . . . . 11 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> S C_ T )
116	115, 59	sseldd 3143	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 2nd ` ( R ` k ) ) e. T )
117		opelxpi 4636	. . . . . . . . . . 11 |- ( ( ( ( G ` k ) + 1 ) e. ( ZZ>= ` C ) /\ ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) e. S ) -> <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
118	44, 60, 117	syl2anc 409	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
119		oveq1 5849	. . . . . . . . . . . 12 |- ( x = ( G ` k ) -> ( x + 1 ) = ( ( G ` k ) + 1 ) )
120	119, 47	opeq12d 3766	. . . . . . . . . . 11 |- ( x = ( G ` k ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F y ) >. )
121	45	opeq2d 3765	. . . . . . . . . . 11 |- ( y = ( 2nd ` ( R ` k ) ) -> <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F y ) >. = <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. )
122	120, 121, 88	ovmpog 5976	. . . . . . . . . 10 |- ( ( ( G ` k ) e. ( ZZ>= ` C ) /\ ( 2nd ` ( R ` k ) ) e. T /\ <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( G ` k ) ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` k ) ) ) = <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. )
123	42, 116, 118, 122	syl3anc 1228	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( ( G ` k ) ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` k ) ) ) = <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. )
124	114, 123	eqtr3id 2213	. . . . . . . 8 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` k ) , ( 2nd ` ( R ` k ) ) >. ) = <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. )
125	113, 124	eqtrd 2198	. . . . . . 7 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( R ` suc k ) = <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. )
126	125	fveq2d 5490	. . . . . 6 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 1st ` ( R ` suc k ) ) = ( 1st ` <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) >. ) )
127	22	ad2antlr 481	. . . . . . 7 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> C e. ZZ )
128	127, 34, 41	frec2uzsucd 10336	. . . . . 6 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( G ` suc k ) = ( ( G ` k ) + 1 ) )
129	62, 126, 128	3eqtr4d 2208	. . . . 5 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 1st ` ( R ` suc k ) ) = ( G ` suc k ) )
130	129	exp31 362	. . . 4 |- ( k e. om -> ( ph -> ( ( 1st ` ( R ` k ) ) = ( G ` k ) -> ( 1st ` ( R ` suc k ) ) = ( G ` suc k ) ) ) )
131	130	a2d 26	. . 3 |- ( k e. om -> ( ( ph -> ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( ph -> ( 1st ` ( R ` suc k ) ) = ( G ` suc k ) ) ) )
132	6, 11, 16, 21, 36, 131	finds 4577	. 2 |- ( N e. om -> ( ph -> ( 1st ` ( R ` N ) ) = ( G ` N ) ) )
133	1, 132	mpcom 36	1 |- ( ph -> ( 1st ` ( R ` N ) ) = ( G ` N ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   C_ wss 3116   (/)c0 3409   <.cop 3579   |-> cmpt 4043   suc csuc 4343   omcom 4567   X. cxp 4602   -->wf 5184   -1-1-onto->wf1o 5187   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   1stc1st 6106   2ndc2nd 6107  freccfrec 6358   1c1 7754   + caddc 7756   ZZcz 9191   ZZ>=cuz 9466

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467

This theorem is referenced by:  frecuzrdgdomlem  10352  frecuzrdgfunlem  10354  frecuzrdgsuctlem  10358