Theorem frecuzrdgtcl 10368

Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10355 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 26-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	|- ( ph -> C e. ZZ )
frec2uz.2	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
frecuzrdgrrn.a	|- ( ph -> A e. S )
frecuzrdgrrn.f	|- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
frecuzrdgrrn.2	|- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
frecuzrdgtcl.3	|- ( ph -> T = ran R )

Assertion

Ref	Expression
frecuzrdgtcl	|- ( ph -> T : ( ZZ>= ` C ) --> S )

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

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

Proof of Theorem frecuzrdgtcl

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

Step	Hyp	Ref	Expression
1		frecuzrdgtcl.3	. . . . . . . . . 10 |- ( ph -> T = ran R )
2	1	eleq2d 2240	. . . . . . . . 9 |- ( ph -> ( z e. T <-> z e. ran R ) )
3		frec2uz.1	. . . . . . . . . . 11 |- ( ph -> C e. ZZ )
4		frec2uz.2	. . . . . . . . . . 11 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
5		frecuzrdgrrn.a	. . . . . . . . . . 11 |- ( ph -> A e. S )
6		frecuzrdgrrn.f	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
7		frecuzrdgrrn.2	. . . . . . . . . . 11 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
8	3, 4, 5, 6, 7	frecuzrdgrcl 10366	. . . . . . . . . 10 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
9		ffn 5347	. . . . . . . . . 10 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> R Fn om )
10		fvelrnb 5544	. . . . . . . . . 10 |- ( R Fn om -> ( z e. ran R <-> E. w e. om ( R ` w ) = z ) )
11	8, 9, 10	3syl 17	. . . . . . . . 9 |- ( ph -> ( z e. ran R <-> E. w e. om ( R ` w ) = z ) )
12	2, 11	bitrd 187	. . . . . . . 8 |- ( ph -> ( z e. T <-> E. w e. om ( R ` w ) = z ) )
13	3, 4, 5, 6, 7	frecuzrdgrrn 10364	. . . . . . . . . 10 |- ( ( ph /\ w e. om ) -> ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) )
14		eleq1 2233	. . . . . . . . . 10 |- ( ( R ` w ) = z -> ( ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) <-> z e. ( ( ZZ>= ` C ) X. S ) ) )
15	13, 14	syl5ibcom 154	. . . . . . . . 9 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. S ) ) )
16	15	rexlimdva 2587	. . . . . . . 8 |- ( ph -> ( E. w e. om ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. S ) ) )
17	12, 16	sylbid 149	. . . . . . 7 |- ( ph -> ( z e. T -> z e. ( ( ZZ>= ` C ) X. S ) ) )
18	17	ssrdv 3153	. . . . . 6 |- ( ph -> T C_ ( ( ZZ>= ` C ) X. S ) )
19		xpss 4719	. . . . . 6 |- ( ( ZZ>= ` C ) X. S ) C_ ( _V X. _V )
20	18, 19	sstrdi 3159	. . . . 5 |- ( ph -> T C_ ( _V X. _V ) )
21		df-rel 4618	. . . . 5 |- ( Rel T <-> T C_ ( _V X. _V ) )
22	20, 21	sylibr 133	. . . 4 |- ( ph -> Rel T )
23	3, 4	frec2uzf1od 10362	. . . . . . . . . . 11 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
24		f1ocnvdm 5760	. . . . . . . . . . 11 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
25	23, 24	sylan 281	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
26	3, 4, 5, 6, 7	frecuzrdgrrn 10364	. . . . . . . . . 10 |- ( ( ph /\ ( `' G ` v ) e. om ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
27	25, 26	syldan 280	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
28		xp2nd 6145	. . . . . . . . 9 |- ( ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S )
29	27, 28	syl 14	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S )
30	1	eleq2d 2240	. . . . . . . . . . . 12 |- ( ph -> ( <. v , z >. e. T <-> <. v , z >. e. ran R ) )
31		fvelrnb 5544	. . . . . . . . . . . . 13 |- ( R Fn om -> ( <. v , z >. e. ran R <-> E. w e. om ( R ` w ) = <. v , z >. ) )
32	8, 9, 31	3syl 17	. . . . . . . . . . . 12 |- ( ph -> ( <. v , z >. e. ran R <-> E. w e. om ( R ` w ) = <. v , z >. ) )
33	30, 32	bitrd 187	. . . . . . . . . . 11 |- ( ph -> ( <. v , z >. e. T <-> E. w e. om ( R ` w ) = <. v , z >. ) )
34	3	adantr 274	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> C e. ZZ )
35	5	adantr 274	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> A e. S )
36	6	adantlr 474	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ w e. om ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
37		simpr 109	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> w e. om )
38	34, 4, 35, 36, 7, 37	frec2uzrdg 10365	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> ( R ` w ) = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
39	38	eqeq1d 2179	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. <-> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. ) )
40		vex 2733	. . . . . . . . . . . . . . . . . . . 20 |- v e. _V
41		vex 2733	. . . . . . . . . . . . . . . . . . . 20 |- z e. _V
42	40, 41	opth2 4225	. . . . . . . . . . . . . . . . . . 19 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. <-> ( ( G ` w ) = v /\ ( 2nd ` ( R ` w ) ) = z ) )
43	42	simplbi 272	. . . . . . . . . . . . . . . . . 18 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. -> ( G ` w ) = v )
44	39, 43	syl6bi 162	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( G ` w ) = v ) )
45		f1ocnvfv 5758	. . . . . . . . . . . . . . . . . 18 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
46	23, 45	sylan 281	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
47	44, 46	syld 45	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( `' G ` v ) = w ) )
48		fveq2 5496	. . . . . . . . . . . . . . . . 17 |- ( ( `' G ` v ) = w -> ( R ` ( `' G ` v ) ) = ( R ` w ) )
49	48	fveq2d 5500	. . . . . . . . . . . . . . . 16 |- ( ( `' G ` v ) = w -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
50	47, 49	syl6 33	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) ) )
51	50	imp 123	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
52	40, 41	op2ndd 6128	. . . . . . . . . . . . . . 15 |- ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` w ) ) = z )
53	52	adantl 275	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` w ) ) = z )
54	51, 53	eqtr2d 2204	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) )
55	54	ex 114	. . . . . . . . . . . 12 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
56	55	rexlimdva 2587	. . . . . . . . . . 11 |- ( ph -> ( E. w e. om ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
57	33, 56	sylbid 149	. . . . . . . . . 10 |- ( ph -> ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
58	57	alrimiv 1867	. . . . . . . . 9 |- ( ph -> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
59	58	adantr 274	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
60		eqeq2 2180	. . . . . . . . . . 11 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( z = w <-> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
61	60	imbi2d 229	. . . . . . . . . 10 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( ( <. v , z >. e. T -> z = w ) <-> ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
62	61	albidv 1817	. . . . . . . . 9 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( A. z ( <. v , z >. e. T -> z = w ) <-> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
63	62	spcegv 2818	. . . . . . . 8 |- ( ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S -> ( A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) -> E. w A. z ( <. v , z >. e. T -> z = w ) ) )
64	29, 59, 63	sylc 62	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> E. w A. z ( <. v , z >. e. T -> z = w ) )
65		nfv 1521	. . . . . . . 8 |- F/ w <. v , z >. e. T
66	65	mo2r 2071	. . . . . . 7 |- ( E. w A. z ( <. v , z >. e. T -> z = w ) -> E* z <. v , z >. e. T )
67	64, 66	syl 14	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> E* z <. v , z >. e. T )
68		dmss 4810	. . . . . . . . . . 11 |- ( T C_ ( ( ZZ>= ` C ) X. S ) -> dom T C_ dom ( ( ZZ>= ` C ) X. S ) )
69	18, 68	syl 14	. . . . . . . . . 10 |- ( ph -> dom T C_ dom ( ( ZZ>= ` C ) X. S ) )
70		dmxpss 5041	. . . . . . . . . 10 |- dom ( ( ZZ>= ` C ) X. S ) C_ ( ZZ>= ` C )
71	69, 70	sstrdi 3159	. . . . . . . . 9 |- ( ph -> dom T C_ ( ZZ>= ` C ) )
72	3	adantr 274	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> C e. ZZ )
73	5	adantr 274	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A e. S )
74	6	adantlr 474	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ v e. ( ZZ>= ` C ) ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
75		simpr 109	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. ( ZZ>= ` C ) )
76	72, 4, 73, 74, 7, 75	frecuzrdglem 10367	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R )
77	1	eleq2d 2240	. . . . . . . . . . . . . 14 |- ( ph -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T <-> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R ) )
78	77	adantr 274	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T <-> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R ) )
79	76, 78	mpbird 166	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T )
80		opeldmg 4816	. . . . . . . . . . . . 13 |- ( ( v e. _V /\ ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S ) -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T -> v e. dom T ) )
81	40, 80	mpan 422	. . . . . . . . . . . 12 |- ( ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T -> v e. dom T ) )
82	29, 79, 81	sylc 62	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. dom T )
83	82	ex 114	. . . . . . . . . 10 |- ( ph -> ( v e. ( ZZ>= ` C ) -> v e. dom T ) )
84	83	ssrdv 3153	. . . . . . . . 9 |- ( ph -> ( ZZ>= ` C ) C_ dom T )
85	71, 84	eqssd 3164	. . . . . . . 8 |- ( ph -> dom T = ( ZZ>= ` C ) )
86	85	eleq2d 2240	. . . . . . 7 |- ( ph -> ( v e. dom T <-> v e. ( ZZ>= ` C ) ) )
87	86	pm5.32i 451	. . . . . 6 |- ( ( ph /\ v e. dom T ) <-> ( ph /\ v e. ( ZZ>= ` C ) ) )
88		df-br 3990	. . . . . . 7 |- ( v T z <-> <. v , z >. e. T )
89	88	mobii 2056	. . . . . 6 |- ( E* z v T z <-> E* z <. v , z >. e. T )
90	67, 87, 89	3imtr4i 200	. . . . 5 |- ( ( ph /\ v e. dom T ) -> E* z v T z )
91	90	ralrimiva 2543	. . . 4 |- ( ph -> A. v e. dom T E* z v T z )
92		dffun7 5225	. . . 4 |- ( Fun T <-> ( Rel T /\ A. v e. dom T E* z v T z ) )
93	22, 91, 92	sylanbrc 415	. . 3 |- ( ph -> Fun T )
94		df-fn 5201	. . 3 |- ( T Fn ( ZZ>= ` C ) <-> ( Fun T /\ dom T = ( ZZ>= ` C ) ) )
95	93, 85, 94	sylanbrc 415	. 2 |- ( ph -> T Fn ( ZZ>= ` C ) )
96		rnss 4841	. . . 4 |- ( T C_ ( ( ZZ>= ` C ) X. S ) -> ran T C_ ran ( ( ZZ>= ` C ) X. S ) )
97	18, 96	syl 14	. . 3 |- ( ph -> ran T C_ ran ( ( ZZ>= ` C ) X. S ) )
98		rnxpss 5042	. . 3 |- ran ( ( ZZ>= ` C ) X. S ) C_ S
99	97, 98	sstrdi 3159	. 2 |- ( ph -> ran T C_ S )
100		df-f 5202	. 2 |- ( T : ( ZZ>= ` C ) --> S <-> ( T Fn ( ZZ>= ` C ) /\ ran T C_ S ) )
101	95, 99, 100	sylanbrc 415	1 |- ( ph -> T : ( ZZ>= ` C ) --> S )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   E.wex 1485   E*wmo 2020   e. wcel 2141   A.wral 2448   E.wrex 2449   _Vcvv 2730   C_ wss 3121   <.cop 3586   class class class wbr 3989   |-> cmpt 4050   omcom 4574   X. cxp 4609   `'ccnv 4610   dom cdm 4611   ran crn 4612   Rel wrel 4616   Fun wfun 5192   Fn wfn 5193   -->wf 5194   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   2ndc2nd 6118  freccfrec 6369   1c1 7775   + caddc 7777   ZZcz 9212   ZZ>=cuz 9487

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488

This theorem is referenced by:  frecuzrdg0  10369  frecuzrdgsuc  10370