Theorem frecuzrdgsuctlem 10860

Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10836 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 29-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 >. )
frecuzrdgsuctlem.g	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
frecuzrdgsuctlem.ran	|- ( ph -> P = ran R )

Assertion

Ref	Expression
frecuzrdgsuctlem	|- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( P ` ( B + 1 ) ) = ( B F ( P ` B ) ) )

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

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

Proof of Theorem frecuzrdgsuctlem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecuzrdgrclt.c	. . . . . 6 |- ( ph -> C e. ZZ )
2		frecuzrdgrclt.a	. . . . . 6 |- ( ph -> A e. S )
3		frecuzrdgrclt.t	. . . . . 6 |- ( ph -> S C_ T )
4		frecuzrdgrclt.f	. . . . . 6 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
5		frecuzrdgrclt.r	. . . . . 6 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
6		frecuzrdgsuctlem.ran	. . . . . 6 |- ( ph -> P = ran R )
7	1, 2, 3, 4, 5, 6	frecuzrdgtclt 10858	. . . . 5 |- ( ph -> P : ( ZZ>= ` C ) --> S )
8	7	adantr 276	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> P : ( ZZ>= ` C ) --> S )
9		ffun 5536	. . . 4 |- ( P : ( ZZ>= ` C ) --> S -> Fun P )
10	8, 9	syl 14	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> Fun P )
11		1st2nd2 6409	. . . . . . . . . . . . . . 15 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
12	11	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
13	12	fveq2d 5699	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ 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 ) >. ) )
14		df-ov 6088	. . . . . . . . . . . . 13 |- ( ( 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 ) >. )
15	13, 14	eqtr4di 2289	. . . . . . . . . . . 12 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) = ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) )
16		xp1st 6399	. . . . . . . . . . . . . 14 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
17	16	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
18	3	ad2antrr 492	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> S C_ T )
19		xp2nd 6400	. . . . . . . . . . . . . . 15 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` z ) e. S )
20	19	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. S )
21	18, 20	sseldd 3249	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. T )
22		peano2uz 9983	. . . . . . . . . . . . . . 15 |- ( ( 1st ` z ) e. ( ZZ>= ` C ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
23	17, 22	syl 14	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
24		oveq2 6093	. . . . . . . . . . . . . . . 16 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) F y ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
25	24	eleq1d 2307	. . . . . . . . . . . . . . 15 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) F y ) e. S <-> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
26		oveq1 6092	. . . . . . . . . . . . . . . . . 18 |- ( x = ( 1st ` z ) -> ( x F y ) = ( ( 1st ` z ) F y ) )
27	26	eleq1d 2307	. . . . . . . . . . . . . . . . 17 |- ( x = ( 1st ` z ) -> ( ( x F y ) e. S <-> ( ( 1st ` z ) F y ) e. S ) )
28	27	ralbidv 2550	. . . . . . . . . . . . . . . 16 |- ( x = ( 1st ` z ) -> ( A. y e. S ( x F y ) e. S <-> A. y e. S ( ( 1st ` z ) F y ) e. S ) )
29	4	ralrimivva 2632	. . . . . . . . . . . . . . . . 17 |- ( ph -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
30	29	ad2antrr 492	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
31	28, 30, 17	rspcdva 2934	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> A. y e. S ( ( 1st ` z ) F y ) e. S )
32	25, 31, 20	rspcdva 2934	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S )
33		opelxpi 4806	. . . . . . . . . . . . . 14 |- ( ( ( ( 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 ) )
34	23, 32, 33	syl2anc 415	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
35		oveq1 6092	. . . . . . . . . . . . . . 15 |- ( x = ( 1st ` z ) -> ( x + 1 ) = ( ( 1st ` z ) + 1 ) )
36	35, 26	opeq12d 3912	. . . . . . . . . . . . . 14 |- ( x = ( 1st ` z ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. )
37	24	opeq2d 3911	. . . . . . . . . . . . . 14 |- ( y = ( 2nd ` z ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
38		eqid 2238	. . . . . . . . . . . . . 14 |- ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) = ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. )
39	36, 37, 38	ovmpog 6223	. . . . . . . . . . . . 13 |- ( ( ( 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 ) ) >. )
40	17, 21, 34, 39	syl3anc 1278	. . . . . . . . . . . 12 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ 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 ) ) >. )
41	15, 40	eqtrd 2271	. . . . . . . . . . 11 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
42	41, 34	eqeltrd 2315	. . . . . . . . . 10 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. ( ( ZZ>= ` C ) X. S ) )
43	42	ralrimiva 2623	. . . . . . . . 9 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> 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 ) )
44		uzid 9936	. . . . . . . . . . . 12 |- ( C e. ZZ -> C e. ( ZZ>= ` C ) )
45	1, 44	syl 14	. . . . . . . . . . 11 |- ( ph -> C e. ( ZZ>= ` C ) )
46		opelxpi 4806	. . . . . . . . . . 11 |- ( ( C e. ( ZZ>= ` C ) /\ A e. S ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
47	45, 2, 46	syl2anc 415	. . . . . . . . . 10 |- ( ph -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
48	47	adantr 276	. . . . . . . . 9 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
49		frecuzrdgsuctlem.g	. . . . . . . . . . 11 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
50	1, 49	frec2uzf1od 10843	. . . . . . . . . 10 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
51		f1ocnvdm 5987	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` B ) e. om )
52	50, 51	sylan 283	. . . . . . . . 9 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` B ) e. om )
53		frecsuc 6678	. . . . . . . . 9 |- ( ( 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 ) /\ ( `' G ` B ) e. om ) -> (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc ( `' G ` B ) ) = ( ( 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 >. ) ` ( `' G ` B ) ) ) )
54	43, 48, 52, 53	syl3anc 1278	. . . . . . . 8 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc ( `' G ` B ) ) = ( ( 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 >. ) ` ( `' G ` B ) ) ) )
55	5	fveq1i 5696	. . . . . . . 8 |- ( R ` suc ( `' G ` B ) ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc ( `' G ` B ) )
56	5	fveq1i 5696	. . . . . . . . 9 |- ( R ` ( `' G ` B ) ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` ( `' G ` B ) )
57	56	fveq2i 5698	. . . . . . . 8 |- ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) = ( ( 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 >. ) ` ( `' G ` B ) ) )
58	54, 55, 57	3eqtr4g 2296	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` suc ( `' G ` B ) ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) )
59	1, 2, 3, 4, 5	frecuzrdgrclt 10852	. . . . . . . . . . . 12 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
60	59	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> R : om --> ( ( ZZ>= ` C ) X. S ) )
61	60, 52	ffvelcdmd 5844	. . . . . . . . . 10 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) )
62		1st2nd2 6409	. . . . . . . . . 10 |- ( ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) -> ( R ` ( `' G ` B ) ) = <. ( 1st ` ( R ` ( `' G ` B ) ) ) , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. )
63	61, 62	syl 14	. . . . . . . . 9 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` B ) ) = <. ( 1st ` ( R ` ( `' G ` B ) ) ) , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. )
64	1	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> C e. ZZ )
65	2	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> A e. S )
66	3	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> S C_ T )
67	4	adantlr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
68	64, 65, 66, 67, 5, 52, 49	frecuzrdgg 10853	. . . . . . . . . . 11 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` B ) ) ) = ( G ` ( `' G ` B ) ) )
69		f1ocnvfv2 5984	. . . . . . . . . . . 12 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ B e. ( ZZ>= ` C ) ) -> ( G ` ( `' G ` B ) ) = B )
70	50, 69	sylan 283	. . . . . . . . . . 11 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( G ` ( `' G ` B ) ) = B )
71	68, 70	eqtrd 2271	. . . . . . . . . 10 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` B ) ) ) = B )
72	71	opeq1d 3910	. . . . . . . . 9 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. ( 1st ` ( R ` ( `' G ` B ) ) ) , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. = <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. )
73	63, 72	eqtrd 2271	. . . . . . . 8 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` B ) ) = <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. )
74	73	fveq2d 5699	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. ) )
75	58, 74	eqtrd 2271	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` suc ( `' G ` B ) ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. ) )
76		df-ov 6088	. . . . . 6 |- ( B ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. )
77	75, 76	eqtr4di 2289	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` suc ( `' G ` B ) ) = ( B ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
78		simpr 110	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> B e. ( ZZ>= ` C ) )
79		xp2nd 6400	. . . . . . . 8 |- ( ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` ( R ` ( `' G ` B ) ) ) e. S )
80	61, 79	syl 14	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` ( `' G ` B ) ) ) e. S )
81	66, 80	sseldd 3249	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` ( `' G ` B ) ) ) e. T )
82		peano2uz 9983	. . . . . . . 8 |- ( B e. ( ZZ>= ` C ) -> ( B + 1 ) e. ( ZZ>= ` C ) )
83	82	adantl 277	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( B + 1 ) e. ( ZZ>= ` C ) )
84	67, 78, 80	caovcld 6243	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) e. S )
85		opelxp 4804	. . . . . . 7 |- ( <. ( B + 1 ) , ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. e. ( ( ZZ>= ` C ) X. S ) <-> ( ( B + 1 ) e. ( ZZ>= ` C ) /\ ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) e. S ) )
86	83, 84, 85	sylanbrc 421	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. ( B + 1 ) , ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
87		oveq1 6092	. . . . . . . 8 |- ( x = B -> ( x + 1 ) = ( B + 1 ) )
88		oveq1 6092	. . . . . . . 8 |- ( x = B -> ( x F y ) = ( B F y ) )
89	87, 88	opeq12d 3912	. . . . . . 7 |- ( x = B -> <. ( x + 1 ) , ( x F y ) >. = <. ( B + 1 ) , ( B F y ) >. )
90		oveq2 6093	. . . . . . . 8 |- ( y = ( 2nd ` ( R ` ( `' G ` B ) ) ) -> ( B F y ) = ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
91	90	opeq2d 3911	. . . . . . 7 |- ( y = ( 2nd ` ( R ` ( `' G ` B ) ) ) -> <. ( B + 1 ) , ( B F y ) >. = <. ( B + 1 ) , ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. )
92	89, 91, 38	ovmpog 6223	. . . . . 6 |- ( ( B e. ( ZZ>= ` C ) /\ ( 2nd ` ( R ` ( `' G ` B ) ) ) e. T /\ <. ( B + 1 ) , ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. e. ( ( ZZ>= ` C ) X. S ) ) -> ( B ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = <. ( B + 1 ) , ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. )
93	78, 81, 86, 92	syl3anc 1278	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( B ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = <. ( B + 1 ) , ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. )
94	77, 93	eqtrd 2271	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` suc ( `' G ` B ) ) = <. ( B + 1 ) , ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. )
95		ffun 5536	. . . . . . 7 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> Fun R )
96	60, 95	syl 14	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> Fun R )
97		peano2 4742	. . . . . . . 8 |- ( ( `' G ` B ) e. om -> suc ( `' G ` B ) e. om )
98	52, 97	syl 14	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> suc ( `' G ` B ) e. om )
99		fdm 5539	. . . . . . . 8 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> dom R = om )
100	60, 99	syl 14	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> dom R = om )
101	98, 100	eleqtrrd 2318	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> suc ( `' G ` B ) e. dom R )
102		fvelrn 5839	. . . . . 6 |- ( ( Fun R /\ suc ( `' G ` B ) e. dom R ) -> ( R ` suc ( `' G ` B ) ) e. ran R )
103	96, 101, 102	syl2anc 415	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` suc ( `' G ` B ) ) e. ran R )
104	6	adantr 276	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> P = ran R )
105	103, 104	eleqtrrd 2318	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` suc ( `' G ` B ) ) e. P )
106	94, 105	eqeltrrd 2316	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. ( B + 1 ) , ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. e. P )
107		funopfv 5740	. . 3 |- ( Fun P -> ( <. ( B + 1 ) , ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. e. P -> ( P ` ( B + 1 ) ) = ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) ) )
108	10, 106, 107	sylc 62	. 2 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( P ` ( B + 1 ) ) = ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
109	52, 100	eleqtrrd 2318	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` B ) e. dom R )
110		fvelrn 5839	. . . . . . 7 |- ( ( Fun R /\ ( `' G ` B ) e. dom R ) -> ( R ` ( `' G ` B ) ) e. ran R )
111	96, 109, 110	syl2anc 415	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` B ) ) e. ran R )
112	111, 104	eleqtrrd 2318	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` B ) ) e. P )
113	73, 112	eqeltrrd 2316	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. P )
114		funopfv 5740	. . . 4 |- ( Fun P -> ( <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. P -> ( P ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
115	10, 113, 114	sylc 62	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( P ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) )
116	115	oveq2d 6101	. 2 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( B F ( P ` B ) ) = ( B F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
117	108, 116	eqtr4d 2274	1 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( P ` ( B + 1 ) ) = ( B F ( P ` B ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   <.cop 3712   |-> cmpt 4192   suc csuc 4510   omcom 4737   X. cxp 4772   `'ccnv 4773   dom cdm 4774   ran crn 4775   Fun wfun 5371   -->wf 5373   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373  freccfrec 6661   1c1 8180   + caddc 8182   ZZcz 9644   ZZ>=cuz 9921

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922

This theorem is used by:  frecuzrdgsuct  10861