Theorem frecuzrdgtcl 10347

Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10334 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 2236	. . . . . . . . 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 10345	. . . . . . . . . 10 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
9		ffn 5337	. . . . . . . . . 10 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> R Fn om )
10		fvelrnb 5534	. . . . . . . . . 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 10343	. . . . . . . . . 10 |- ( ( ph /\ w e. om ) -> ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) )
14		eleq1 2229	. . . . . . . . . 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 2583	. . . . . . . 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 3148	. . . . . 6 |- ( ph -> T C_ ( ( ZZ>= ` C ) X. S ) )
19		xpss 4712	. . . . . 6 |- ( ( ZZ>= ` C ) X. S ) C_ ( _V X. _V )
20	18, 19	sstrdi 3154	. . . . 5 |- ( ph -> T C_ ( _V X. _V ) )
21		df-rel 4611	. . . . 5 |- ( Rel T <-> T C_ ( _V X. _V ) )
22	20, 21	sylibr 133	. . . 4 |- ( ph -> Rel T )
23	3, 4	frec2uzf1od 10341	. . . . . . . . . . 11 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
24		f1ocnvdm 5749	. . . . . . . . . . 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 10343	. . . . . . . . . 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 6134	. . . . . . . . 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 2236	. . . . . . . . . . . 12 |- ( ph -> ( <. v , z >. e. T <-> <. v , z >. e. ran R ) )
31		fvelrnb 5534	. . . . . . . . . . . . 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 469	. . . . . . . . . . . . . . . . . . . 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 10344	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> ( R ` w ) = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
39	38	eqeq1d 2174	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. <-> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. ) )
40		vex 2729	. . . . . . . . . . . . . . . . . . . 20 |- v e. _V
41		vex 2729	. . . . . . . . . . . . . . . . . . . 20 |- z e. _V
42	40, 41	opth2 4218	. . . . . . . . . . . . . . . . . . 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 5747	. . . . . . . . . . . . . . . . . 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 5486	. . . . . . . . . . . . . . . . 17 |- ( ( `' G ` v ) = w -> ( R ` ( `' G ` v ) ) = ( R ` w ) )
49	48	fveq2d 5490	. . . . . . . . . . . . . . . 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 6117	. . . . . . . . . . . . . . 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 2199	. . . . . . . . . . . . 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 2583	. . . . . . . . . . 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 1862	. . . . . . . . 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 2175	. . . . . . . . . . 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 1812	. . . . . . . . 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 2814	. . . . . . . 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 1516	. . . . . . . 8 |- F/ w <. v , z >. e. T
66	65	mo2r 2066	. . . . . . 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 4803	. . . . . . . . . . 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 5034	. . . . . . . . . 10 |- dom ( ( ZZ>= ` C ) X. S ) C_ ( ZZ>= ` C )
71	69, 70	sstrdi 3154	. . . . . . . . 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 469	. . . . . . . . . . . . . 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 10346	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R )
77	1	eleq2d 2236	. . . . . . . . . . . . . 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 4809	. . . . . . . . . . . . 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 421	. . . . . . . . . . . 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 3148	. . . . . . . . 9 |- ( ph -> ( ZZ>= ` C ) C_ dom T )
85	71, 84	eqssd 3159	. . . . . . . 8 |- ( ph -> dom T = ( ZZ>= ` C ) )
86	85	eleq2d 2236	. . . . . . 7 |- ( ph -> ( v e. dom T <-> v e. ( ZZ>= ` C ) ) )
87	86	pm5.32i 450	. . . . . 6 |- ( ( ph /\ v e. dom T ) <-> ( ph /\ v e. ( ZZ>= ` C ) ) )
88		df-br 3983	. . . . . . 7 |- ( v T z <-> <. v , z >. e. T )
89	88	mobii 2051	. . . . . 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 2539	. . . 4 |- ( ph -> A. v e. dom T E* z v T z )
92		dffun7 5215	. . . 4 |- ( Fun T <-> ( Rel T /\ A. v e. dom T E* z v T z ) )
93	22, 91, 92	sylanbrc 414	. . 3 |- ( ph -> Fun T )
94		df-fn 5191	. . 3 |- ( T Fn ( ZZ>= ` C ) <-> ( Fun T /\ dom T = ( ZZ>= ` C ) ) )
95	93, 85, 94	sylanbrc 414	. 2 |- ( ph -> T Fn ( ZZ>= ` C ) )
96		rnss 4834	. . . 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 5035	. . 3 |- ran ( ( ZZ>= ` C ) X. S ) C_ S
99	97, 98	sstrdi 3154	. 2 |- ( ph -> ran T C_ S )
100		df-f 5192	. 2 |- ( T : ( ZZ>= ` C ) --> S <-> ( T Fn ( ZZ>= ` C ) /\ ran T C_ S ) )
101	95, 99, 100	sylanbrc 414	1 |- ( ph -> T : ( ZZ>= ` C ) --> S )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   E.wex 1480   E*wmo 2015   e. wcel 2136   A.wral 2444   E.wrex 2445   _Vcvv 2726   C_ wss 3116   <.cop 3579   class class class wbr 3982   |-> cmpt 4043   omcom 4567   X. cxp 4602   `'ccnv 4603   dom cdm 4604   ran crn 4605   Rel wrel 4609   Fun wfun 5182   Fn wfn 5183   -->wf 5184   -1-1-onto->wf1o 5187   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   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:  frecuzrdg0  10348  frecuzrdgsuc  10349