Theorem frecuzrdgtcl 10414

Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10401 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 2247	. . . . . . . . 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 10412	. . . . . . . . . 10 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
9		ffn 5367	. . . . . . . . . 10 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> R Fn om )
10		fvelrnb 5565	. . . . . . . . . 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 188	. . . . . . . 8 |- ( ph -> ( z e. T <-> E. w e. om ( R ` w ) = z ) )
13	3, 4, 5, 6, 7	frecuzrdgrrn 10410	. . . . . . . . . 10 |- ( ( ph /\ w e. om ) -> ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) )
14		eleq1 2240	. . . . . . . . . 10 |- ( ( R ` w ) = z -> ( ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) <-> z e. ( ( ZZ>= ` C ) X. S ) ) )
15	13, 14	syl5ibcom 155	. . . . . . . . 9 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. S ) ) )
16	15	rexlimdva 2594	. . . . . . . 8 |- ( ph -> ( E. w e. om ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. S ) ) )
17	12, 16	sylbid 150	. . . . . . 7 |- ( ph -> ( z e. T -> z e. ( ( ZZ>= ` C ) X. S ) ) )
18	17	ssrdv 3163	. . . . . 6 |- ( ph -> T C_ ( ( ZZ>= ` C ) X. S ) )
19		xpss 4736	. . . . . 6 |- ( ( ZZ>= ` C ) X. S ) C_ ( _V X. _V )
20	18, 19	sstrdi 3169	. . . . 5 |- ( ph -> T C_ ( _V X. _V ) )
21		df-rel 4635	. . . . 5 |- ( Rel T <-> T C_ ( _V X. _V ) )
22	20, 21	sylibr 134	. . . 4 |- ( ph -> Rel T )
23	3, 4	frec2uzf1od 10408	. . . . . . . . . . 11 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
24		f1ocnvdm 5784	. . . . . . . . . . 11 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
25	23, 24	sylan 283	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
26	3, 4, 5, 6, 7	frecuzrdgrrn 10410	. . . . . . . . . 10 |- ( ( ph /\ ( `' G ` v ) e. om ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
27	25, 26	syldan 282	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
28		xp2nd 6169	. . . . . . . . 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 2247	. . . . . . . . . . . 12 |- ( ph -> ( <. v , z >. e. T <-> <. v , z >. e. ran R ) )
31		fvelrnb 5565	. . . . . . . . . . . . 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 188	. . . . . . . . . . 11 |- ( ph -> ( <. v , z >. e. T <-> E. w e. om ( R ` w ) = <. v , z >. ) )
34	3	adantr 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> C e. ZZ )
35	5	adantr 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> A e. S )
36	6	adantlr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ w e. om ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
37		simpr 110	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> w e. om )
38	34, 4, 35, 36, 7, 37	frec2uzrdg 10411	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> ( R ` w ) = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
39	38	eqeq1d 2186	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. <-> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. ) )
40		vex 2742	. . . . . . . . . . . . . . . . . . . 20 |- v e. _V
41		vex 2742	. . . . . . . . . . . . . . . . . . . 20 |- z e. _V
42	40, 41	opth2 4242	. . . . . . . . . . . . . . . . . . 19 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. <-> ( ( G ` w ) = v /\ ( 2nd ` ( R ` w ) ) = z ) )
43	42	simplbi 274	. . . . . . . . . . . . . . . . . 18 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. -> ( G ` w ) = v )
44	39, 43	biimtrdi 163	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( G ` w ) = v ) )
45		f1ocnvfv 5782	. . . . . . . . . . . . . . . . . 18 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
46	23, 45	sylan 283	. . . . . . . . . . . . . . . . 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 5517	. . . . . . . . . . . . . . . . 17 |- ( ( `' G ` v ) = w -> ( R ` ( `' G ` v ) ) = ( R ` w ) )
49	48	fveq2d 5521	. . . . . . . . . . . . . . . 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 124	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
52	40, 41	op2ndd 6152	. . . . . . . . . . . . . . 15 |- ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` w ) ) = z )
53	52	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` w ) ) = z )
54	51, 53	eqtr2d 2211	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) )
55	54	ex 115	. . . . . . . . . . . 12 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
56	55	rexlimdva 2594	. . . . . . . . . . 11 |- ( ph -> ( E. w e. om ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
57	33, 56	sylbid 150	. . . . . . . . . 10 |- ( ph -> ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
58	57	alrimiv 1874	. . . . . . . . 9 |- ( ph -> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
59	58	adantr 276	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A. z ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
60		eqeq2 2187	. . . . . . . . . . 11 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( z = w <-> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
61	60	imbi2d 230	. . . . . . . . . 10 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( ( <. v , z >. e. T -> z = w ) <-> ( <. v , z >. e. T -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
62	61	albidv 1824	. . . . . . . . 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 2827	. . . . . . . 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 1528	. . . . . . . 8 |- F/ w <. v , z >. e. T
66	65	mo2r 2078	. . . . . . 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 4828	. . . . . . . . . . 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 5061	. . . . . . . . . 10 |- dom ( ( ZZ>= ` C ) X. S ) C_ ( ZZ>= ` C )
71	69, 70	sstrdi 3169	. . . . . . . . 9 |- ( ph -> dom T C_ ( ZZ>= ` C ) )
72	3	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> C e. ZZ )
73	5	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A e. S )
74	6	adantlr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ v e. ( ZZ>= ` C ) ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
75		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. ( ZZ>= ` C ) )
76	72, 4, 73, 74, 7, 75	frecuzrdglem 10413	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R )
77	1	eleq2d 2247	. . . . . . . . . . . . . 14 |- ( ph -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T <-> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R ) )
78	77	adantr 276	. . . . . . . . . . . . 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 167	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. T )
80		opeldmg 4834	. . . . . . . . . . . . 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 424	. . . . . . . . . . . 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 115	. . . . . . . . . 10 |- ( ph -> ( v e. ( ZZ>= ` C ) -> v e. dom T ) )
84	83	ssrdv 3163	. . . . . . . . 9 |- ( ph -> ( ZZ>= ` C ) C_ dom T )
85	71, 84	eqssd 3174	. . . . . . . 8 |- ( ph -> dom T = ( ZZ>= ` C ) )
86	85	eleq2d 2247	. . . . . . 7 |- ( ph -> ( v e. dom T <-> v e. ( ZZ>= ` C ) ) )
87	86	pm5.32i 454	. . . . . 6 |- ( ( ph /\ v e. dom T ) <-> ( ph /\ v e. ( ZZ>= ` C ) ) )
88		df-br 4006	. . . . . . 7 |- ( v T z <-> <. v , z >. e. T )
89	88	mobii 2063	. . . . . 6 |- ( E* z v T z <-> E* z <. v , z >. e. T )
90	67, 87, 89	3imtr4i 201	. . . . 5 |- ( ( ph /\ v e. dom T ) -> E* z v T z )
91	90	ralrimiva 2550	. . . 4 |- ( ph -> A. v e. dom T E* z v T z )
92		dffun7 5245	. . . 4 |- ( Fun T <-> ( Rel T /\ A. v e. dom T E* z v T z ) )
93	22, 91, 92	sylanbrc 417	. . 3 |- ( ph -> Fun T )
94		df-fn 5221	. . 3 |- ( T Fn ( ZZ>= ` C ) <-> ( Fun T /\ dom T = ( ZZ>= ` C ) ) )
95	93, 85, 94	sylanbrc 417	. 2 |- ( ph -> T Fn ( ZZ>= ` C ) )
96		rnss 4859	. . . 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 5062	. . 3 |- ran ( ( ZZ>= ` C ) X. S ) C_ S
99	97, 98	sstrdi 3169	. 2 |- ( ph -> ran T C_ S )
100		df-f 5222	. 2 |- ( T : ( ZZ>= ` C ) --> S <-> ( T Fn ( ZZ>= ` C ) /\ ran T C_ S ) )
101	95, 99, 100	sylanbrc 417	1 |- ( ph -> T : ( ZZ>= ` C ) --> S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1492   E*wmo 2027   e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2739   C_ wss 3131   <.cop 3597   class class class wbr 4005   |-> cmpt 4066   omcom 4591   X. cxp 4626   `'ccnv 4627   dom cdm 4628   ran crn 4629   Rel wrel 4633   Fun wfun 5212   Fn wfn 5213   -->wf 5214   -1-1-onto->wf1o 5217   ` cfv 5218  (class class class)co 5877   e. cmpo 5879   2ndc2nd 6142  freccfrec 6393   1c1 7814   + caddc 7816   ZZcz 9255   ZZ>=cuz 9530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531

This theorem is referenced by:  frecuzrdg0  10415  frecuzrdgsuc  10416