Theorem frecuzrdgfn 9198

Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9185 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 )
uzrdg.s	|- ( ph -> S e. V )
uzrdg.a	|- ( ph -> A e. S )
uzrdg.f	|- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
uzrdg.2	|- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
frecuzrdgfn.3	|- ( ph -> T = ran R )

Assertion

Ref	Expression
frecuzrdgfn	|- ( ph -> T Fn ( ZZ>= ` C ) )

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)   V( x, y)

Proof of Theorem frecuzrdgfn

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

Step	Hyp	Ref	Expression
1		frecuzrdgfn.3	. . . . . . . . 9 |- ( ph -> T = ran R )
2	1	eleq2d 2107	. . . . . . . 8 |- ( ph -> ( z e. T <-> z e. ran R ) )
3		frec2uz.1	. . . . . . . . . 10 |- ( ph -> C e. ZZ )
4		frec2uz.2	. . . . . . . . . 10 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
5		uzrdg.s	. . . . . . . . . 10 |- ( ph -> S e. V )
6		uzrdg.a	. . . . . . . . . 10 |- ( ph -> A e. S )
7		uzrdg.f	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
8		uzrdg.2	. . . . . . . . . 10 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
9	3, 4, 5, 6, 7, 8	frecuzrdgrom 9196	. . . . . . . . 9 |- ( ph -> R Fn om )
10		fvelrnb 5221	. . . . . . . . 9 |- ( R Fn om -> ( z e. ran R <-> E. w e. om ( R ` w ) = z ) )
11	9, 10	syl 14	. . . . . . . 8 |- ( ph -> ( z e. ran R <-> E. w e. om ( R ` w ) = z ) )
12	2, 11	bitrd 177	. . . . . . 7 |- ( ph -> ( z e. T <-> E. w e. om ( R ` w ) = z ) )
13	3, 4, 5, 6, 7, 8	frecuzrdgrrn 9194	. . . . . . . . 9 |- ( ( ph /\ w e. om ) -> ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) )
14		eleq1 2100	. . . . . . . . 9 |- ( ( R ` w ) = z -> ( ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) <-> z e. ( ( ZZ>= ` C ) X. S ) ) )
15	13, 14	syl5ibcom 144	. . . . . . . 8 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. S ) ) )
16	15	rexlimdva 2433	. . . . . . 7 |- ( ph -> ( E. w e. om ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. S ) ) )
17	12, 16	sylbid 139	. . . . . 6 |- ( ph -> ( z e. T -> z e. ( ( ZZ>= ` C ) X. S ) ) )
18	17	ssrdv 2951	. . . . 5 |- ( ph -> T C_ ( ( ZZ>= ` C ) X. S ) )
19		xpss 4446	. . . . 5 |- ( ( ZZ>= ` C ) X. S ) C_ ( _V X. _V )
20	18, 19	syl6ss 2957	. . . 4 |- ( ph -> T C_ ( _V X. _V ) )
21		df-rel 4352	. . . 4 |- ( Rel T <-> T C_ ( _V X. _V ) )
22	20, 21	sylibr 137	. . 3 |- ( ph -> Rel T )
23	3, 4	frec2uzf1od 9192	. . . . . . . . . 10 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
24		f1ocnvdm 5421	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
25	23, 24	sylan 267	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
26	3, 4, 5, 6, 7, 8	frecuzrdgrrn 9194	. . . . . . . . 9 |- ( ( ph /\ ( `' G ` v ) e. om ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
27	25, 26	syldan 266	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
28		xp2nd 5793	. . . . . . . 8 |- ( ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S )
29	27, 28	syl 14	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S )
30	1	eleq2d 2107	. . . . . . . . . . 11 |- ( ph -> ( <. v , z >. e. T <-> <. v , z >. e. ran R ) )
31		fvelrnb 5221	. . . . . . . . . . . 12 |- ( R Fn om -> ( <. v , z >. e. ran R <-> E. w e. om ( R ` w ) = <. v , z >. ) )
32	9, 31	syl 14	. . . . . . . . . . 11 |- ( ph -> ( <. v , z >. e. ran R <-> E. w e. om ( R ` w ) = <. v , z >. ) )
33	30, 32	bitrd 177	. . . . . . . . . 10 |- ( ph -> ( <. v , z >. e. T <-> E. w e. om ( R ` w ) = <. v , z >. ) )
34	3	adantr 261	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> C e. ZZ )
35	5	adantr 261	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> S e. V )
36	6	adantr 261	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> A e. S )
37	7	adantlr 446	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ w e. om ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
38		simpr 103	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> w e. om )
39	34, 4, 35, 36, 37, 8, 38	frec2uzrdg 9195	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w e. om ) -> ( R ` w ) = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
40	39	eqeq1d 2048	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. <-> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. ) )
41		vex 2560	. . . . . . . . . . . . . . . . . . 19 |- v e. _V
42		vex 2560	. . . . . . . . . . . . . . . . . . 19 |- z e. _V
43	41, 42	opth2 3977	. . . . . . . . . . . . . . . . . 18 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. <-> ( ( G ` w ) = v /\ ( 2nd ` ( R ` w ) ) = z ) )
44	43	simplbi 259	. . . . . . . . . . . . . . . . 17 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. -> ( G ` w ) = v )
45	40, 44	syl6bi 152	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( G ` w ) = v ) )
46		f1ocnvfv 5419	. . . . . . . . . . . . . . . . 17 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
47	23, 46	sylan 267	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
48	45, 47	syld 40	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( `' G ` v ) = w ) )
49		fveq2 5178	. . . . . . . . . . . . . . . 16 |- ( ( `' G ` v ) = w -> ( R ` ( `' G ` v ) ) = ( R ` w ) )
50	49	fveq2d 5182	. . . . . . . . . . . . . . 15 |- ( ( `' G ` v ) = w -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
51	48, 50	syl6 29	. . . . . . . . . . . . . 14 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) ) )
52	51	imp 115	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
53	41, 42	op2ndd 5776	. . . . . . . . . . . . . 14 |- ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` w ) ) = z )
54	53	adantl 262	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` w ) ) = z )
55	52, 54	eqtr2d 2073	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) )
56	55	ex 108	. . . . . . . . . . 11 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
57	56	rexlimdva 2433	. . . . . . . . . 10 |- ( ph -> ( E. w e. om ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
58	33, 57	sylbid 139	. . . . . . . . 9 |- ( ph -> ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
59	58	alrimiv 1754	. . . . . . . 8 |- ( ph -> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
60	59	adantr 261	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
61		eqeq2 2049	. . . . . . . . . 10 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( z = w <-> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
62	61	imbi2d 219	. . . . . . . . 9 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( ( <. v , z >. e. T -> z = w ) <-> ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
63	62	albidv 1705	. . . . . . . 8 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( A. z ( <. v , z >. e. T -> z = w ) <-> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
64	63	spcegv 2641	. . . . . . 7 |- ( ( 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 ) ) )
65	29, 60, 64	sylc 56	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> E. w A. z ( <. v , z >. e. T -> z = w ) )
66		nfv 1421	. . . . . . 7 |- F/ w <. v , z >. e. T
67	66	mo2r 1952	. . . . . 6 |- ( E. w A. z ( <. v , z >. e. T -> z = w ) -> E* z <. v , z >. e. T )
68	65, 67	syl 14	. . . . 5 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> E* z <. v , z >. e. T )
69		dmss 4534	. . . . . . . . . 10 |- ( T C_ ( ( ZZ>= ` C ) X. S ) -> dom T C_ dom ( ( ZZ>= ` C ) X. S ) )
70	18, 69	syl 14	. . . . . . . . 9 |- ( ph -> dom T C_ dom ( ( ZZ>= ` C ) X. S ) )
71		dmxpss 4753	. . . . . . . . 9 |- dom ( ( ZZ>= ` C ) X. S ) C_ ( ZZ>= ` C )
72	70, 71	syl6ss 2957	. . . . . . . 8 |- ( ph -> dom T C_ ( ZZ>= ` C ) )
73	3	adantr 261	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> C e. ZZ )
74	5	adantr 261	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> S e. V )
75	6	adantr 261	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A e. S )
76	7	adantlr 446	. . . . . . . . . . . . 13 |- ( ( ( ph /\ v e. ( ZZ>= ` C ) ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
77		simpr 103	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. ( ZZ>= ` C ) )
78	73, 4, 74, 75, 76, 8, 77	frecuzrdglem 9197	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R )
79	1	eleq2d 2107	. . . . . . . . . . . . 13 |- ( ph -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T <-> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R ) )
80	79	adantr 261	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T <-> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R ) )
81	78, 80	mpbird 156	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T )
82		opeldmg 4540	. . . . . . . . . . . 12 |- ( ( v e. _V /\ ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S ) -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T -> v e. dom T ) )
83	41, 82	mpan 400	. . . . . . . . . . 11 |- ( ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T -> v e. dom T ) )
84	29, 81, 83	sylc 56	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. dom T )
85	84	ex 108	. . . . . . . . 9 |- ( ph -> ( v e. ( ZZ>= ` C ) -> v e. dom T ) )
86	85	ssrdv 2951	. . . . . . . 8 |- ( ph -> ( ZZ>= ` C ) C_ dom T )
87	72, 86	eqssd 2962	. . . . . . 7 |- ( ph -> dom T = ( ZZ>= ` C ) )
88	87	eleq2d 2107	. . . . . 6 |- ( ph -> ( v e. dom T <-> v e. ( ZZ>= ` C ) ) )
89	88	pm5.32i 427	. . . . 5 |- ( ( ph /\ v e. dom T ) <-> ( ph /\ v e. ( ZZ>= ` C ) ) )
90		df-br 3765	. . . . . 6 |- ( v T z <-> <. v , z >. e. T )
91	90	mobii 1937	. . . . 5 |- ( E* z v T z <-> E* z <. v , z >. e. T )
92	68, 89, 91	3imtr4i 190	. . . 4 |- ( ( ph /\ v e. dom T ) -> E* z v T z )
93	92	ralrimiva 2392	. . 3 |- ( ph -> A. v e. dom T E* z v T z )
94		dffun7 4928	. . 3 |- ( Fun T <-> ( Rel T /\ A. v e. dom T E* z v T z ) )
95	22, 93, 94	sylanbrc 394	. 2 |- ( ph -> Fun T )
96		df-fn 4905	. 2 |- ( T Fn ( ZZ>= ` C ) <-> ( Fun T /\ dom T = ( ZZ>= ` C ) ) )
97	95, 87, 96	sylanbrc 394	1 |- ( ph -> T Fn ( ZZ>= ` C ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   E*wmo 1901   A.wral 2306   E.wrex 2307   _Vcvv 2557   C_ wss 2917   <.cop 3378   class class class wbr 3764   |-> cmpt 3818   omcom 4313   X. cxp 4343   `'ccnv 4344   dom cdm 4345   ran crn 4346   Rel wrel 4350   Fun wfun 4896   Fn wfn 4897   -1-1-onto->wf1o 4901   ` cfv 4902  (class class class)co 5512   |-> cmpt2 5514   2ndc2nd 5766  freccfrec 5977   1c1 6890   + caddc 6892   ZZcz 8245   ZZ>=cuz 8473

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474

This theorem is referenced by:  frecuzrdgcl  9199  frecuzrdg0  9200  frecuzrdgsuc  9201  iseqfn  9221