Theorem frecuzrdgg 10182

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 5414	. . . . . 6 |- ( w = (/) -> ( R ` w ) = ( R ` (/) ) )
3	2	fveq2d 5418	. . . . 5 |- ( w = (/) -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` (/) ) ) )
4		fveq2 5414	. . . . 5 |- ( w = (/) -> ( G ` w ) = ( G ` (/) ) )
5	3, 4	eqeq12d 2152	. . . 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 5414	. . . . . 6 |- ( w = k -> ( R ` w ) = ( R ` k ) )
8	7	fveq2d 5418	. . . . 5 |- ( w = k -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` k ) ) )
9		fveq2 5414	. . . . 5 |- ( w = k -> ( G ` w ) = ( G ` k ) )
10	8, 9	eqeq12d 2152	. . . 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 5414	. . . . . 6 |- ( w = suc k -> ( R ` w ) = ( R ` suc k ) )
13	12	fveq2d 5418	. . . . 5 |- ( w = suc k -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` suc k ) ) )
14		fveq2 5414	. . . . 5 |- ( w = suc k -> ( G ` w ) = ( G ` suc k ) )
15	13, 14	eqeq12d 2152	. . . 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 5414	. . . . . 6 |- ( w = N -> ( R ` w ) = ( R ` N ) )
18	17	fveq2d 5418	. . . . 5 |- ( w = N -> ( 1st ` ( R ` w ) ) = ( 1st ` ( R ` N ) ) )
19		fveq2 5414	. . . . 5 |- ( w = N -> ( G ` w ) = ( G ` N ) )
20	18, 19	eqeq12d 2152	. . . 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 6041	. . . . 5 |- ( ( C e. ZZ /\ A e. S ) -> ( 1st ` <. C , A >. ) = C )
25	22, 23, 24	syl2anc 408	. . . 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 5415	. . . . . 6 |- ( R ` (/) ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` (/) )
28		opexg 4145	. . . . . . . 8 |- ( ( C e. ZZ /\ A e. S ) -> <. C , A >. e. _V )
29		frec0g 6287	. . . . . . . 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 408	. . . . . 6 |- ( ph -> (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` (/) ) = <. C , A >. )
32	27, 31	syl5eq 2182	. . . . 5 |- ( ph -> ( R ` (/) ) = <. C , A >. )
33	32	fveq2d 5418	. . . 4 |- ( ph -> ( 1st ` ( R ` (/) ) ) = ( 1st ` <. C , A >. ) )
34		frecuzrdgg.g	. . . . 5 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
35	22, 34	frec2uz0d 10165	. . . 4 |- ( ph -> ( G ` (/) ) = C )
36	25, 33, 35	3eqtr4d 2180	. . 3 |- ( ph -> ( 1st ` ( R ` (/) ) ) = ( G ` (/) ) )
37	22, 34	frec2uzf1od 10172	. . . . . . . . . . 11 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
38		f1of 5360	. . . . . . . . . . 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 480	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> G : om --> ( ZZ>= ` C ) )
41		simpll 518	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> k e. om )
42	40, 41	ffvelrnd 5549	. . . . . . . 8 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( G ` k ) e. ( ZZ>= ` C ) )
43		peano2uz 9371	. . . . . . . 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 5775	. . . . . . . . 9 |- ( y = ( 2nd ` ( R ` k ) ) -> ( ( G ` k ) F y ) = ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) )
46	45	eleq1d 2206	. . . . . . . 8 |- ( y = ( 2nd ` ( R ` k ) ) -> ( ( ( G ` k ) F y ) e. S <-> ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) e. S ) )
47		oveq1 5774	. . . . . . . . . . 11 |- ( x = ( G ` k ) -> ( x F y ) = ( ( G ` k ) F y ) )
48	47	eleq1d 2206	. . . . . . . . . 10 |- ( x = ( G ` k ) -> ( ( x F y ) e. S <-> ( ( G ` k ) F y ) e. S ) )
49	48	ralbidv 2435	. . . . . . . . 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 2512	. . . . . . . . . 10 |- ( ph -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
52	51	ad2antlr 480	. . . . . . . . 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 2789	. . . . . . . 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 10181	. . . . . . . . . . 11 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
56	55	ad2antlr 480	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> R : om --> ( ( ZZ>= ` C ) X. S ) )
57	56, 41	ffvelrnd 5549	. . . . . . . . 9 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( R ` k ) e. ( ( ZZ>= ` C ) X. S ) )
58		xp2nd 6057	. . . . . . . . 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 2789	. . . . . . 7 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( ( G ` k ) F ( 2nd ` ( R ` k ) ) ) e. S )
61		op1stg 6041	. . . . . . 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 408	. . . . . 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 6066	. . . . . . . . . . . . . . . 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 5418	. . . . . . . . . . . . . 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 5770	. . . . . . . . . . . . . . . 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 6056	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> S C_ T )
70		xp2nd 6057	. . . . . . . . . . . . . . . . . . 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 3093	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. T )
73		peano2uz 9371	. . . . . . . . . . . . . . . . . . 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 5775	. . . . . . . . . . . . . . . . . . . 20 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) F y ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
76	75	eleq1d 2206	. . . . . . . . . . . . . . . . . . 19 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) F y ) e. S <-> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
77		oveq1 5774	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x = ( 1st ` z ) -> ( x F y ) = ( ( 1st ` z ) F y ) )
78	77	eleq1d 2206	. . . . . . . . . . . . . . . . . . . . 21 |- ( x = ( 1st ` z ) -> ( ( x F y ) e. S <-> ( ( 1st ` z ) F y ) e. S ) )
79	78	ralbidv 2435	. . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . . 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 4566	. . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . 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 5774	. . . . . . . . . . . . . . . . . . 19 |- ( x = ( 1st ` z ) -> ( x + 1 ) = ( ( 1st ` z ) + 1 ) )
86	85, 77	opeq12d 3708	. . . . . . . . . . . . . . . . . 18 |- ( x = ( 1st ` z ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. )
87	75	opeq2d 3707	. . . . . . . . . . . . . . . . . 18 |- ( y = ( 2nd ` z ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
88		eqid 2137	. . . . . . . . . . . . . . . . . 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 5898	. . . . . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . . . . 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	syl5eqr 2184	. . . . . . . . . . . . . . 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 2214	. . . . . . . . . . . . . 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 2214	. . . . . . . . . . . . 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 2503	. . . . . . . . . . . 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 9333	. . . . . . . . . . . . . . 15 |- ( C e. ZZ -> C e. ( ZZ>= ` C ) )
96	22, 95	syl 14	. . . . . . . . . . . . . 14 |- ( ph -> C e. ( ZZ>= ` C ) )
97		opelxpi 4566	. . . . . . . . . . . . . 14 |- ( ( C e. ( ZZ>= ` C ) /\ A e. S ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
98	96, 23, 97	syl2anc 408	. . . . . . . . . . . . 13 |- ( ph -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
99	98	ad2antlr 480	. . . . . . . . . . . 12 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
100		frecsuc 6297	. . . . . . . . . . . 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 1216	. . . . . . . . . . 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 5415	. . . . . . . . . . 11 |- ( R ` suc k ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc k )
103	26	fveq1i 5415	. . . . . . . . . . . 12 |- ( R ` k ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` k )
104	103	fveq2i 5417	. . . . . . . . . . 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 2195	. . . . . . . . . 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 6066	. . . . . . . . . . . 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 5418	. . . . . . . . . 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 2170	. . . . . . . . 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 3706	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> <. ( 1st ` ( R ` k ) ) , ( 2nd ` ( R ` k ) ) >. = <. ( G ` k ) , ( 2nd ` ( R ` k ) ) >. )
112	111	fveq2d 5418	. . . . . . . . 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 2170	. . . . . . . 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 5770	. . . . . . . . 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 480	. . . . . . . . . . 11 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> S C_ T )
116	115, 59	sseldd 3093	. . . . . . . . . 10 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 2nd ` ( R ` k ) ) e. T )
117		opelxpi 4566	. . . . . . . . . . 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 408	. . . . . . . . . 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 5774	. . . . . . . . . . . 12 |- ( x = ( G ` k ) -> ( x + 1 ) = ( ( G ` k ) + 1 ) )
120	119, 47	opeq12d 3708	. . . . . . . . . . 11 |- ( x = ( G ` k ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( G ` k ) + 1 ) , ( ( G ` k ) F y ) >. )
121	45	opeq2d 3707	. . . . . . . . . . 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 5898	. . . . . . . . . 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 1216	. . . . . . . . 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	syl5eqr 2184	. . . . . . . 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 2170	. . . . . . 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 5418	. . . . . 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 480	. . . . . . 7 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> C e. ZZ )
128	127, 34, 41	frec2uzsucd 10167	. . . . . 6 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( G ` suc k ) = ( ( G ` k ) + 1 ) )
129	62, 126, 128	3eqtr4d 2180	. . . . 5 |- ( ( ( k e. om /\ ph ) /\ ( 1st ` ( R ` k ) ) = ( G ` k ) ) -> ( 1st ` ( R ` suc k ) ) = ( G ` suc k ) )
130	129	exp31 361	. . . 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 4509	. 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 1331   e. wcel 1480   A.wral 2414   _Vcvv 2681   C_ wss 3066   (/)c0 3358   <.cop 3525   |-> cmpt 3984   suc csuc 4282   omcom 4499   X. cxp 4532   -->wf 5114   -1-1-onto->wf1o 5117   ` cfv 5118  (class class class)co 5767   e. cmpo 5769   1stc1st 6029   2ndc2nd 6030  freccfrec 6280   1c1 7614   + caddc 7616   ZZcz 9047   ZZ>=cuz 9319

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320

This theorem is referenced by:  frecuzrdgdomlem  10183  frecuzrdgfunlem  10185  frecuzrdgsuctlem  10189