Theorem frecuzrdgtcl 9546

Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9533 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 2152	. . . . . . . . 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 9544	. . . . . . . . . 10 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
9		ffn 5097	. . . . . . . . . 10 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> R Fn om )
10		fvelrnb 5273	. . . . . . . . . 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 186	. . . . . . . 8 |- ( ph -> ( z e. T <-> E. w e. om ( R ` w ) = z ) )
13	3, 4, 5, 6, 7	frecuzrdgrrn 9542	. . . . . . . . . 10 |- ( ( ph /\ w e. om ) -> ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) )
14		eleq1 2145	. . . . . . . . . 10 |- ( ( R ` w ) = z -> ( ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) <-> z e. ( ( ZZ>= ` C ) X. S ) ) )
15	13, 14	syl5ibcom 153	. . . . . . . . 9 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. S ) ) )
16	15	rexlimdva 2482	. . . . . . . 8 |- ( ph -> ( E. w e. om ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. S ) ) )
17	12, 16	sylbid 148	. . . . . . 7 |- ( ph -> ( z e. T -> z e. ( ( ZZ>= ` C ) X. S ) ) )
18	17	ssrdv 3014	. . . . . 6 |- ( ph -> T C_ ( ( ZZ>= ` C ) X. S ) )
19		xpss 4494	. . . . . 6 |- ( ( ZZ>= ` C ) X. S ) C_ ( _V X. _V )
20	18, 19	syl6ss 3020	. . . . 5 |- ( ph -> T C_ ( _V X. _V ) )
21		df-rel 4398	. . . . 5 |- ( Rel T <-> T C_ ( _V X. _V ) )
22	20, 21	sylibr 132	. . . 4 |- ( ph -> Rel T )
23	3, 4	frec2uzf1od 9540	. . . . . . . . . . 11 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
24		f1ocnvdm 5472	. . . . . . . . . . 11 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
25	23, 24	sylan 277	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
26	3, 4, 5, 6, 7	frecuzrdgrrn 9542	. . . . . . . . . 10 |- ( ( ph /\ ( `' G ` v ) e. om ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
27	25, 26	syldan 276	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
28		xp2nd 5844	. . . . . . . . 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 2152	. . . . . . . . . . . 12 |- ( ph -> ( <. v , z >. e. T <-> <. v , z >. e. ran R ) )
31		fvelrnb 5273	. . . . . . . . . . . . 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 186	. . . . . . . . . . 11 |- ( ph -> ( <. v , z >. e. T <-> E. w e. om ( R ` w ) = <. v , z >. ) )
34	3	adantr 270	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> C e. ZZ )
35	5	adantr 270	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> A e. S )
36	6	adantlr 461	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ w e. om ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
37		simpr 108	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> w e. om )
38	34, 4, 35, 36, 7, 37	frec2uzrdg 9543	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> ( R ` w ) = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
39	38	eqeq1d 2091	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. <-> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. ) )
40		vex 2613	. . . . . . . . . . . . . . . . . . . 20 |- v e. _V
41		vex 2613	. . . . . . . . . . . . . . . . . . . 20 |- z e. _V
42	40, 41	opth2 4023	. . . . . . . . . . . . . . . . . . 19 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. <-> ( ( G ` w ) = v /\ ( 2nd ` ( R ` w ) ) = z ) )
43	42	simplbi 268	. . . . . . . . . . . . . . . . . 18 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. -> ( G ` w ) = v )
44	39, 43	syl6bi 161	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( G ` w ) = v ) )
45		f1ocnvfv 5470	. . . . . . . . . . . . . . . . . 18 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
46	23, 45	sylan 277	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
47	44, 46	syld 44	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( `' G ` v ) = w ) )
48		fveq2 5229	. . . . . . . . . . . . . . . . 17 |- ( ( `' G ` v ) = w -> ( R ` ( `' G ` v ) ) = ( R ` w ) )
49	48	fveq2d 5233	. . . . . . . . . . . . . . . 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 122	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
52	40, 41	op2ndd 5827	. . . . . . . . . . . . . . 15 |- ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` w ) ) = z )
53	52	adantl 271	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` w ) ) = z )
54	51, 53	eqtr2d 2116	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) )
55	54	ex 113	. . . . . . . . . . . 12 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
56	55	rexlimdva 2482	. . . . . . . . . . 11 |- ( ph -> ( E. w e. om ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
57	33, 56	sylbid 148	. . . . . . . . . 10 |- ( ph -> ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
58	57	alrimiv 1797	. . . . . . . . 9 |- ( ph -> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
59	58	adantr 270	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
60		eqeq2 2092	. . . . . . . . . . 11 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( z = w <-> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
61	60	imbi2d 228	. . . . . . . . . 10 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( ( <. v , z >. e. T -> z = w ) <-> ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
62	61	albidv 1747	. . . . . . . . 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 2695	. . . . . . . 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 61	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> E. w A. z ( <. v , z >. e. T -> z = w ) )
65		nfv 1462	. . . . . . . 8 |- F/ w <. v , z >. e. T
66	65	mo2r 1995	. . . . . . 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 4582	. . . . . . . . . . 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 4803	. . . . . . . . . 10 |- dom ( ( ZZ>= ` C ) X. S ) C_ ( ZZ>= ` C )
71	69, 70	syl6ss 3020	. . . . . . . . 9 |- ( ph -> dom T C_ ( ZZ>= ` C ) )
72	3	adantr 270	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> C e. ZZ )
73	5	adantr 270	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A e. S )
74	6	adantlr 461	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ v e. ( ZZ>= ` C ) ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
75		simpr 108	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. ( ZZ>= ` C ) )
76	72, 4, 73, 74, 7, 75	frecuzrdglem 9545	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R )
77	1	eleq2d 2152	. . . . . . . . . . . . . 14 |- ( ph -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T <-> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R ) )
78	77	adantr 270	. . . . . . . . . . . . 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 165	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T )
80		opeldmg 4588	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 12 |- ( ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T -> v e. dom T ) )
82	29, 79, 81	sylc 61	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. dom T )
83	82	ex 113	. . . . . . . . . 10 |- ( ph -> ( v e. ( ZZ>= ` C ) -> v e. dom T ) )
84	83	ssrdv 3014	. . . . . . . . 9 |- ( ph -> ( ZZ>= ` C ) C_ dom T )
85	71, 84	eqssd 3025	. . . . . . . 8 |- ( ph -> dom T = ( ZZ>= ` C ) )
86	85	eleq2d 2152	. . . . . . 7 |- ( ph -> ( v e. dom T <-> v e. ( ZZ>= ` C ) ) )
87	86	pm5.32i 442	. . . . . 6 |- ( ( ph /\ v e. dom T ) <-> ( ph /\ v e. ( ZZ>= ` C ) ) )
88		df-br 3806	. . . . . . 7 |- ( v T z <-> <. v , z >. e. T )
89	88	mobii 1980	. . . . . 6 |- ( E* z v T z <-> E* z <. v , z >. e. T )
90	67, 87, 89	3imtr4i 199	. . . . 5 |- ( ( ph /\ v e. dom T ) -> E* z v T z )
91	90	ralrimiva 2439	. . . 4 |- ( ph -> A. v e. dom T E* z v T z )
92		dffun7 4978	. . . 4 |- ( Fun T <-> ( Rel T /\ A. v e. dom T E* z v T z ) )
93	22, 91, 92	sylanbrc 408	. . 3 |- ( ph -> Fun T )
94		df-fn 4955	. . 3 |- ( T Fn ( ZZ>= ` C ) <-> ( Fun T /\ dom T = ( ZZ>= ` C ) ) )
95	93, 85, 94	sylanbrc 408	. 2 |- ( ph -> T Fn ( ZZ>= ` C ) )
96		rnss 4612	. . . 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 4804	. . 3 |- ran ( ( ZZ>= ` C ) X. S ) C_ S
99	97, 98	syl6ss 3020	. 2 |- ( ph -> ran T C_ S )
100		df-f 4956	. 2 |- ( T : ( ZZ>= ` C ) --> S <-> ( T Fn ( ZZ>= ` C ) /\ ran T C_ S ) )
101	95, 99, 100	sylanbrc 408	1 |- ( ph -> T : ( ZZ>= ` C ) --> S )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1283   = wceq 1285   E.wex 1422   e. wcel 1434   E*wmo 1944   A.wral 2353   E.wrex 2354   _Vcvv 2610   C_ wss 2982   <.cop 3419   class class class wbr 3805   |-> cmpt 3859   omcom 4359   X. cxp 4389   `'ccnv 4390   dom cdm 4391   ran crn 4392   Rel wrel 4396   Fun wfun 4946   Fn wfn 4947   -->wf 4948   -1-1-onto->wf1o 4951   ` cfv 4952  (class class class)co 5563   |-> cmpt2 5565   2ndc2nd 5817  freccfrec 6059   1c1 7096   + caddc 7098   ZZcz 8484   ZZ>=cuz 8752

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753

This theorem is referenced by:  frecuzrdg0  9547  frecuzrdgsuc  9548  iseqfcl  9587  iseqcl  9589