Theorem frecuzrdgfn 9414

Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9401 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 2148	. . . . . . . 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 9412	. . . . . . . . 9 |- ( ph -> R Fn om )
10		fvelrnb 5242	. . . . . . . . 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 186	. . . . . . 7 |- ( ph -> ( z e. T <-> E. w e. om ( R ` w ) = z ) )
13	3, 4, 5, 6, 7, 8	frecuzrdgrrn 9410	. . . . . . . . 9 |- ( ( ph /\ w e. om ) -> ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) )
14		eleq1 2141	. . . . . . . . 9 |- ( ( R ` w ) = z -> ( ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) <-> z e. ( ( ZZ>= ` C ) X. S ) ) )
15	13, 14	syl5ibcom 153	. . . . . . . 8 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. S ) ) )
16	15	rexlimdva 2477	. . . . . . 7 |- ( ph -> ( E. w e. om ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. S ) ) )
17	12, 16	sylbid 148	. . . . . 6 |- ( ph -> ( z e. T -> z e. ( ( ZZ>= ` C ) X. S ) ) )
18	17	ssrdv 3005	. . . . 5 |- ( ph -> T C_ ( ( ZZ>= ` C ) X. S ) )
19		xpss 4464	. . . . 5 |- ( ( ZZ>= ` C ) X. S ) C_ ( _V X. _V )
20	18, 19	syl6ss 3011	. . . 4 |- ( ph -> T C_ ( _V X. _V ) )
21		df-rel 4370	. . . 4 |- ( Rel T <-> T C_ ( _V X. _V ) )
22	20, 21	sylibr 132	. . 3 |- ( ph -> Rel T )
23	3, 4	frec2uzf1od 9408	. . . . . . . . . 10 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
24		f1ocnvdm 5441	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
25	23, 24	sylan 277	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
26	3, 4, 5, 6, 7, 8	frecuzrdgrrn 9410	. . . . . . . . 9 |- ( ( ph /\ ( `' G ` v ) e. om ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
27	25, 26	syldan 276	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
28		xp2nd 5813	. . . . . . . 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 2148	. . . . . . . . . . 11 |- ( ph -> ( <. v , z >. e. T <-> <. v , z >. e. ran R ) )
31		fvelrnb 5242	. . . . . . . . . . . 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 186	. . . . . . . . . 10 |- ( ph -> ( <. v , z >. e. T <-> E. w e. om ( R ` w ) = <. v , z >. ) )
34	3	adantr 270	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> C e. ZZ )
35	5	adantr 270	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> S e. V )
36	6	adantr 270	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> A e. S )
37	7	adantlr 460	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ w e. om ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
38		simpr 108	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> w e. om )
39	34, 4, 35, 36, 37, 8, 38	frec2uzrdg 9411	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w e. om ) -> ( R ` w ) = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
40	39	eqeq1d 2089	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. <-> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. ) )
41		vex 2604	. . . . . . . . . . . . . . . . . . 19 |- v e. _V
42		vex 2604	. . . . . . . . . . . . . . . . . . 19 |- z e. _V
43	41, 42	opth2 3995	. . . . . . . . . . . . . . . . . 18 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. <-> ( ( G ` w ) = v /\ ( 2nd ` ( R ` w ) ) = z ) )
44	43	simplbi 268	. . . . . . . . . . . . . . . . 17 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. -> ( G ` w ) = v )
45	40, 44	syl6bi 161	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( G ` w ) = v ) )
46		f1ocnvfv 5439	. . . . . . . . . . . . . . . . 17 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
47	23, 46	sylan 277	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
48	45, 47	syld 44	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( `' G ` v ) = w ) )
49		fveq2 5198	. . . . . . . . . . . . . . . 16 |- ( ( `' G ` v ) = w -> ( R ` ( `' G ` v ) ) = ( R ` w ) )
50	49	fveq2d 5202	. . . . . . . . . . . . . . 15 |- ( ( `' G ` v ) = w -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
51	48, 50	syl6 33	. . . . . . . . . . . . . 14 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) ) )
52	51	imp 122	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
53	41, 42	op2ndd 5796	. . . . . . . . . . . . . 14 |- ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` w ) ) = z )
54	53	adantl 271	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` w ) ) = z )
55	52, 54	eqtr2d 2114	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) )
56	55	ex 113	. . . . . . . . . . 11 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
57	56	rexlimdva 2477	. . . . . . . . . 10 |- ( ph -> ( E. w e. om ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
58	33, 57	sylbid 148	. . . . . . . . 9 |- ( ph -> ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
59	58	alrimiv 1795	. . . . . . . 8 |- ( ph -> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
60	59	adantr 270	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
61		eqeq2 2090	. . . . . . . . . 10 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( z = w <-> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
62	61	imbi2d 228	. . . . . . . . 9 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( ( <. v , z >. e. T -> z = w ) <-> ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
63	62	albidv 1745	. . . . . . . 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 2686	. . . . . . 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 61	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> E. w A. z ( <. v , z >. e. T -> z = w ) )
66		nfv 1461	. . . . . . 7 |- F/ w <. v , z >. e. T
67	66	mo2r 1993	. . . . . 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 4552	. . . . . . . . . 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 4773	. . . . . . . . 9 |- dom ( ( ZZ>= ` C ) X. S ) C_ ( ZZ>= ` C )
72	70, 71	syl6ss 3011	. . . . . . . 8 |- ( ph -> dom T C_ ( ZZ>= ` C ) )
73	3	adantr 270	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> C e. ZZ )
74	5	adantr 270	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> S e. V )
75	6	adantr 270	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A e. S )
76	7	adantlr 460	. . . . . . . . . . . . 13 |- ( ( ( ph /\ v e. ( ZZ>= ` C ) ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
77		simpr 108	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. ( ZZ>= ` C ) )
78	73, 4, 74, 75, 76, 8, 77	frecuzrdglem 9413	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R )
79	1	eleq2d 2148	. . . . . . . . . . . . 13 |- ( ph -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T <-> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R ) )
80	79	adantr 270	. . . . . . . . . . . 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 165	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T )
82		opeldmg 4558	. . . . . . . . . . . 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 414	. . . . . . . . . . 11 |- ( ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T -> v e. dom T ) )
84	29, 81, 83	sylc 61	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. dom T )
85	84	ex 113	. . . . . . . . 9 |- ( ph -> ( v e. ( ZZ>= ` C ) -> v e. dom T ) )
86	85	ssrdv 3005	. . . . . . . 8 |- ( ph -> ( ZZ>= ` C ) C_ dom T )
87	72, 86	eqssd 3016	. . . . . . 7 |- ( ph -> dom T = ( ZZ>= ` C ) )
88	87	eleq2d 2148	. . . . . 6 |- ( ph -> ( v e. dom T <-> v e. ( ZZ>= ` C ) ) )
89	88	pm5.32i 441	. . . . 5 |- ( ( ph /\ v e. dom T ) <-> ( ph /\ v e. ( ZZ>= ` C ) ) )
90		df-br 3786	. . . . . 6 |- ( v T z <-> <. v , z >. e. T )
91	90	mobii 1978	. . . . 5 |- ( E* z v T z <-> E* z <. v , z >. e. T )
92	68, 89, 91	3imtr4i 199	. . . 4 |- ( ( ph /\ v e. dom T ) -> E* z v T z )
93	92	ralrimiva 2434	. . 3 |- ( ph -> A. v e. dom T E* z v T z )
94		dffun7 4948	. . 3 |- ( Fun T <-> ( Rel T /\ A. v e. dom T E* z v T z ) )
95	22, 93, 94	sylanbrc 408	. 2 |- ( ph -> Fun T )
96		df-fn 4925	. 2 |- ( T Fn ( ZZ>= ` C ) <-> ( Fun T /\ dom T = ( ZZ>= ` C ) ) )
97	95, 87, 96	sylanbrc 408	1 |- ( ph -> T Fn ( ZZ>= ` C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   E*wmo 1942   A.wral 2348   E.wrex 2349   _Vcvv 2601   C_ wss 2973   <.cop 3401   class class class wbr 3785   |-> cmpt 3839   omcom 4331   X. cxp 4361   `'ccnv 4362   dom cdm 4363   ran crn 4364   Rel wrel 4368   Fun wfun 4916   Fn wfn 4917   -1-1-onto->wf1o 4921   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534   2ndc2nd 5786  freccfrec 6000   1c1 6982   + caddc 6984   ZZcz 8351   ZZ>=cuz 8619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620

This theorem is referenced by:  frecuzrdgcl  9415  frecuzrdg0  9416  frecuzrdgsuc  9417  iseqfn  9441