Theorem frecuzrdgg 10598

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 5599	. . . . . 6 |- ( w = (/) -> ( R ` w ) = ( R ` (/) ) )
3	2	fveq2d 5603	. . . . 5 |- ( w = (/) -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` (/) ) ) )
4		fveq2 5599	. . . . 5 |- ( w = (/) -> ( G ` w ) = ( G ` (/) ) )
5	3, 4	eqeq12d 2222	. . . 4 |- ( w = (/) -> ( ( 1st ` ( R ` w ) ) = ( G ` w ) <-> ( 1st ` ( R ` (/) ) ) = ( G ` (/) ) ) )
6	5	imbi2d 230	. . 3 |- ( w = (/) -> ( ( ph -> ( 1st ` ( R ` w ) ) = ( G ` w ) ) <-> ( ph -> ( 1st ` ( R ` (/) ) ) = ( G ` (/) ) ) ) )
7		fveq2 5599	. . . . . 6 |- ( w = k -> ( R ` w ) = ( R ` k ) )
8	7	fveq2d 5603	. . . . 5 |- ( w = k -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` k ) ) )
9		fveq2 5599	. . . . 5 |- ( w = k -> ( G ` w ) = ( G ` k ) )
10	8, 9	eqeq12d 2222	. . . 4 |- ( w = k -> ( ( 1st ` ( R ` w ) ) = ( G ` w ) <-> ( 1st ` ( R ` k ) ) = ( G ` k ) ) )
11	10	imbi2d 230	. . 3 |- ( w = k -> ( ( ph -> ( 1st ` ( R ` w ) ) = ( G ` w ) ) <-> ( ph -> ( 1st ` ( R ` k ) ) = ( G ` k ) ) ) )
12		fveq2 5599	. . . . . 6 |- ( w = suc k -> ( R ` w ) = ( R ` suc k ) )
13	12	fveq2d 5603	. . . . 5 |- ( w = suc k -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` suc k ) ) )
14		fveq2 5599	. . . . 5 |- ( w = suc k -> ( G ` w ) = ( G ` suc k ) )
15	13, 14	eqeq12d 2222	. . . 4 |- ( w = suc k -> ( ( 1st ` ( R ` w ) ) = ( G ` w ) <-> ( 1st ` ( R ` suc k ) ) = ( G ` suc k ) ) )
16	15	imbi2d 230	. . 3 |- ( w = suc k -> ( ( ph -> ( 1st ` ( R ` w ) ) = ( G ` w ) ) <-> ( ph -> ( 1st ` ( R ` suc k ) ) = ( G ` suc k ) ) ) )
17		fveq2 5599	. . . . . 6 |- ( w = N -> ( R ` w ) = ( R ` N ) )
18	17	fveq2d 5603	. . . . 5 |- ( w = N -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` N ) ) )
19		fveq2 5599	. . . . 5 |- ( w = N -> ( G ` w ) = ( G ` N ) )
20	18, 19	eqeq12d 2222	. . . 4 |- ( w = N -> ( ( 1st ` ( R ` w ) ) = ( G ` w ) <-> ( 1st ` ( R ` N ) ) = ( G ` N ) ) )
21	20	imbi2d 230	. . 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 6259	. . . . 5 |- ( ( C e. ZZ /\ A e. S ) -> ( 1st ` <. C , A >. ) = C )
25	22, 23, 24	syl2anc 411	. . . 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 5600	. . . . . 6 |- ( R ` (/) ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` (/) )
28		opexg 4290	. . . . . . . 8 |- ( ( C e. ZZ /\ A e. S ) -> <. C , A >. e. _V )
29		frec0g 6506	. . . . . . . 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 411	. . . . . 6 |- ( ph -> (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` (/) ) = <. C , A >. )
32	27, 31	eqtrid 2252	. . . . 5 |- ( ph -> ( R ` (/) ) = <. C , A >. )
33	32	fveq2d 5603	. . . 4 |- ( ph -> ( 1st ` ( R ` (/) ) ) = ( 1st ` <. C , A >. ) )
34		frecuzrdgg.g	. . . . 5 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
35	22, 34	frec2uz0d 10581	. . . 4 |- ( ph -> ( G ` (/) ) = C )
36	25, 33, 35	3eqtr4d 2250	. . 3 |- ( ph -> ( 1st ` ( R ` (/) ) ) = ( G ` (/) ) )
37	22, 34	frec2uzf1od 10588	. . . . . . . . . . 11 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
38		f1of 5544	. . . . . . . . . . 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 489	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> G : om --> ( ZZ>= ` C ) )
41		simpll 527	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> k e. om )
42	40, 41	ffvelcdmd 5739	. . . . . . . 8 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( G ` k ) e. ( ZZ>= ` C ) )
43		peano2uz 9739	. . . . . . . 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 5975	. . . . . . . . 9 |- ( y = ( 2nd ` ( R ` k ) ) -> ( ( G ` k ) F y ) = ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) )
46	45	eleq1d 2276	. . . . . . . 8 |- ( y = ( 2nd ` ( R ` k ) ) -> ( ( ( G ` k ) F y ) e. S <-> ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) e. S ) )
47		oveq1 5974	. . . . . . . . . . 11 |- ( x = ( G ` k ) -> ( x F y ) = ( ( G ` k ) F y ) )
48	47	eleq1d 2276	. . . . . . . . . 10 |- ( x = ( G ` k ) -> ( ( x F y ) e. S <-> ( ( G ` k ) F y ) e. S ) )
49	48	ralbidv 2508	. . . . . . . . 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 2590	. . . . . . . . . 10 |- ( ph -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
52	51	ad2antlr 489	. . . . . . . . 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 2889	. . . . . . . 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 10597	. . . . . . . . . . 11 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
56	55	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> R : om --> ( ( ZZ>= ` C ) X. S ) )
57	56, 41	ffvelcdmd 5739	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( R ` k ) e. ( ( ZZ>= ` C ) X. S ) )
58		xp2nd 6275	. . . . . . . . 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 2889	. . . . . . 7 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) e. S )
61		op1stg 6259	. . . . . . 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 411	. . . . . 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 6284	. . . . . . . . . . . . . . . 16 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
64	63	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
65	64	fveq2d 5603	. . . . . . . . . . . . . 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 5970	. . . . . . . . . . . . . . . 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 6274	. . . . . . . . . . . . . . . . . 18 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
68	67	adantl 277	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
69	54	ad3antlr 493	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> S C_ T )
70		xp2nd 6275	. . . . . . . . . . . . . . . . . . 19 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` z ) e. S )
71	70	adantl 277	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. S )
72	69, 71	sseldd 3202	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. T )
73		peano2uz 9739	. . . . . . . . . . . . . . . . . . 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 5975	. . . . . . . . . . . . . . . . . . . 20 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) F y ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
76	75	eleq1d 2276	. . . . . . . . . . . . . . . . . . 19 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) F y ) e. S <-> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
77		oveq1 5974	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x = ( 1st ` z ) -> ( x F y ) = ( ( 1st ` z ) F y ) )
78	77	eleq1d 2276	. . . . . . . . . . . . . . . . . . . . 21 |- ( x = ( 1st ` z ) -> ( ( x F y ) e. S <-> ( ( 1st ` z ) F y ) e. S ) )
79	78	ralbidv 2508	. . . . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . . . 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 2889	. . . . . . . . . . . . . . . . . . 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 2889	. . . . . . . . . . . . . . . . . 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 4725	. . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . 19 |- ( x = ( 1st ` z ) -> ( x + 1 ) = ( ( 1st ` z ) + 1 ) )
86	85, 77	opeq12d 3841	. . . . . . . . . . . . . . . . . 18 |- ( x = ( 1st ` z ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. )
87	75	opeq2d 3840	. . . . . . . . . . . . . . . . . 18 |- ( y = ( 2nd ` z ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
88		eqid 2207	. . . . . . . . . . . . . . . . . 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 6103	. . . . . . . . . . . . . . . . 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 1250	. . . . . . . . . . . . . . . 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 2254	. . . . . . . . . . . . . . 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 2284	. . . . . . . . . . . . . 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 2284	. . . . . . . . . . . . 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 2581	. . . . . . . . . . . 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 9697	. . . . . . . . . . . . . . 15 |- ( C e. ZZ -> C e. ( ZZ>= ` C ) )
96	22, 95	syl 14	. . . . . . . . . . . . . 14 |- ( ph -> C e. ( ZZ>= ` C ) )
97		opelxpi 4725	. . . . . . . . . . . . . 14 |- ( ( C e. ( ZZ>= ` C ) /\ A e. S ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
98	96, 23, 97	syl2anc 411	. . . . . . . . . . . . 13 |- ( ph -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
99	98	ad2antlr 489	. . . . . . . . . . . 12 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
100		frecsuc 6516	. . . . . . . . . . . 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 1250	. . . . . . . . . . 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 5600	. . . . . . . . . . 11 |- ( R ` suc k ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc k )
103	26	fveq1i 5600	. . . . . . . . . . . 12 |- ( R ` k ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` k )
104	103	fveq2i 5602	. . . . . . . . . . 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 2265	. . . . . . . . . 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 6284	. . . . . . . . . . . 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 5603	. . . . . . . . . 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 2240	. . . . . . . . 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 110	. . . . . . . . . . 11 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 1st ` ( R ` k ) ) = ( G ` k ) )
111	110	opeq1d 3839	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> <. ( 1st ` ( R ` k ) ) , ( 2nd ` ( R ` k ) ) >. = <. ( G ` k ) , ( 2nd ` ( R ` k ) ) >. )
112	111	fveq2d 5603	. . . . . . . . 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 2240	. . . . . . . 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 5970	. . . . . . . . 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 489	. . . . . . . . . . 11 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> S C_ T )
116	115, 59	sseldd 3202	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 2nd ` ( R ` k ) ) e. T )
117		opelxpi 4725	. . . . . . . . . . 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 411	. . . . . . . . . 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 5974	. . . . . . . . . . . 12 |- ( x = ( G ` k ) -> ( x + 1 ) = ( ( G ` k ) + 1 ) )
120	119, 47	opeq12d 3841	. . . . . . . . . . 11 |- ( x = ( G ` k ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F y ) >. )
121	45	opeq2d 3840	. . . . . . . . . . 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 6103	. . . . . . . . . 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 1250	. . . . . . . . 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 2254	. . . . . . . 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 2240	. . . . . . 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 5603	. . . . . 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 489	. . . . . . 7 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> C e. ZZ )
128	127, 34, 41	frec2uzsucd 10583	. . . . . 6 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( G ` suc k ) = ( ( G ` k ) + 1 ) )
129	62, 126, 128	3eqtr4d 2250	. . . . 5 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 1st ` ( R ` suc k ) ) = ( G ` suc k ) )
130	129	exp31 364	. . . 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 4666	. 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 104   = wceq 1373   e. wcel 2178   A.wral 2486   _Vcvv 2776   C_ wss 3174   (/)c0 3468   <.cop 3646   |-> cmpt 4121   suc csuc 4430   omcom 4656   X. cxp 4691   -->wf 5286   -1-1-onto->wf1o 5289   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   1stc1st 6247   2ndc2nd 6248  freccfrec 6499   1c1 7961   + caddc 7963   ZZcz 9407   ZZ>=cuz 9683

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684

This theorem is referenced by:  frecuzrdgdomlem  10599  frecuzrdgfunlem  10601  frecuzrdgsuctlem  10605