Theorem frecuzrdgfunlem 10785

Description: The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)

Hypotheses

Ref	Expression
frecuzrdgrclt.c	|- ( ph -> C e. ZZ )
frecuzrdgrclt.a	|- ( ph -> A e. S )
frecuzrdgrclt.t	|- ( ph -> S C_ T )
frecuzrdgrclt.f	|- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
frecuzrdgrclt.r	|- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
frecuzrdgfunlem.g	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )

Assertion

Ref	Expression
frecuzrdgfunlem	|- ( ph -> Fun ran R )

Distinct variable groups:   x, C, y   x, F, y   x, S, y   x, T, y   ph, x, y   x, G, y   x, R, y

Allowed substitution hints:   A( x, y)

Proof of Theorem frecuzrdgfunlem

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

Step	Hyp	Ref	Expression
1		frecuzrdgrclt.c	. . . . . 6 |- ( ph -> C e. ZZ )
2		frecuzrdgrclt.a	. . . . . 6 |- ( ph -> A e. S )
3		frecuzrdgrclt.t	. . . . . 6 |- ( ph -> S C_ T )
4		frecuzrdgrclt.f	. . . . . 6 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
5		frecuzrdgrclt.r	. . . . . 6 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
6	1, 2, 3, 4, 5	frecuzrdgrclt 10781	. . . . 5 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
7		frn 5519	. . . . 5 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> ran R C_ ( ( ZZ>= ` C ) X. S ) )
8	6, 7	syl 14	. . . 4 |- ( ph -> ran R C_ ( ( ZZ>= ` C ) X. S ) )
9		xpss 4860	. . . 4 |- ( ( ZZ>= ` C ) X. S ) C_ ( _V X. _V )
10	8, 9	sstrdi 3252	. . 3 |- ( ph -> ran R C_ ( _V X. _V ) )
11		df-rel 4758	. . 3 |- ( Rel ran R <-> ran R C_ ( _V X. _V ) )
12	10, 11	sylibr 134	. 2 |- ( ph -> Rel ran R )
13		frecuzrdgfunlem.g	. . . . . . . . . 10 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
14	1, 13	frec2uzf1od 10772	. . . . . . . . 9 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
15		f1ocnvdm 5956	. . . . . . . . 9 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
16	14, 15	sylan 283	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
17	6	ffvelcdmda 5814	. . . . . . . 8 |- ( ( ph /\ ( `' G ` v ) e. om ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
18	16, 17	syldan 282	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
19		xp2nd 6362	. . . . . . 7 |- ( ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S )
20	18, 19	syl 14	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S )
21		ffn 5510	. . . . . . . . . 10 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> R Fn om )
22		fvelrnb 5726	. . . . . . . . . 10 |- ( R Fn om -> ( <. v , z >. e. ran R <-> E. w e. om ( R ` w ) = <. v , z >. ) )
23	6, 21, 22	3syl 17	. . . . . . . . 9 |- ( ph -> ( <. v , z >. e. ran R <-> E. w e. om ( R ` w ) = <. v , z >. ) )
24	6	ffvelcdmda 5814	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) )
25		1st2nd2 6371	. . . . . . . . . . . . . . . . . . 19 |- ( ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) -> ( R ` w ) = <. ( 1st ` ( R ` w ) ) , ( 2nd ` ( R ` w ) ) >. )
26	24, 25	syl 14	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w e. om ) -> ( R ` w ) = <. ( 1st ` ( R ` w ) ) , ( 2nd ` ( R ` w ) ) >. )
27	1	adantr 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> C e. ZZ )
28	2	adantr 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> A e. S )
29	3	adantr 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> S C_ T )
30	4	adantlr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ w e. om ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
31		simpr 110	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> w e. om )
32	27, 28, 29, 30, 5, 31, 13	frecuzrdgg 10782	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> ( 1st ` ( R ` w ) ) = ( G ` w ) )
33	32	opeq1d 3891	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w e. om ) -> <. ( 1st ` ( R ` w ) ) , ( 2nd ` ( R ` w ) ) >. = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
34	26, 33	eqtrd 2267	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. om ) -> ( R ` w ) = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
35	34	eqeq1d 2243	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. <-> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. ) )
36		vex 2818	. . . . . . . . . . . . . . . . . 18 |- v e. _V
37		vex 2818	. . . . . . . . . . . . . . . . . 18 |- z e. _V
38	36, 37	opth2 4358	. . . . . . . . . . . . . . . . 17 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. <-> ( ( G ` w ) = v /\ ( 2nd ` ( R ` w ) ) = z ) )
39	38	simplbi 274	. . . . . . . . . . . . . . . 16 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. -> ( G ` w ) = v )
40	35, 39	biimtrdi 163	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( G ` w ) = v ) )
41		f1ocnvfv 5954	. . . . . . . . . . . . . . . 16 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
42	14, 41	sylan 283	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
43	40, 42	syld 45	. . . . . . . . . . . . . 14 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( `' G ` v ) = w ) )
44		fveq2 5672	. . . . . . . . . . . . . . 15 |- ( ( `' G ` v ) = w -> ( R ` ( `' G ` v ) ) = ( R ` w ) )
45	44	fveq2d 5676	. . . . . . . . . . . . . 14 |- ( ( `' G ` v ) = w -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
46	43, 45	syl6 33	. . . . . . . . . . . . 13 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) ) )
47	46	imp 124	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
48	36, 37	op2ndd 6345	. . . . . . . . . . . . 13 |- ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` w ) ) = z )
49	48	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` w ) ) = z )
50	47, 49	eqtr2d 2268	. . . . . . . . . . 11 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) )
51	50	ex 115	. . . . . . . . . 10 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
52	51	rexlimdva 2662	. . . . . . . . 9 |- ( ph -> ( E. w e. om ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
53	23, 52	sylbid 150	. . . . . . . 8 |- ( ph -> ( <. v , z >. e. ran R -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
54	53	alrimiv 1923	. . . . . . 7 |- ( ph -> A. z ( <. v , z >. e. ran R -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
55	54	adantr 276	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A. z ( <. v , z >. e. ran R -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
56		eqeq2 2244	. . . . . . . . 9 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( z = w <-> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
57	56	imbi2d 230	. . . . . . . 8 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( ( <. v , z >. e. ran R -> z = w ) <-> ( <. v , z >. e. ran R -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
58	57	albidv 1873	. . . . . . 7 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( A. z ( <. v , z >. e. ran R -> z = w ) <-> A. z ( <. v , z >. e. ran R -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
59	58	spcegv 2907	. . . . . 6 |- ( ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S -> ( A. z ( <. v , z >. e. ran R -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) -> E. w A. z ( <. v , z >. e. ran R -> z = w ) ) )
60	20, 55, 59	sylc 62	. . . . 5 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> E. w A. z ( <. v , z >. e. ran R -> z = w ) )
61		nfv 1577	. . . . . 6 |- F/ w <. v , z >. e. ran R
62	61	mo2r 2135	. . . . 5 |- ( E. w A. z ( <. v , z >. e. ran R -> z = w ) -> E* z <. v , z >. e. ran R )
63	60, 62	syl 14	. . . 4 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> E* z <. v , z >. e. ran R )
64	1, 2, 3, 4, 5	frecuzrdgdom 10784	. . . . . 6 |- ( ph -> dom ran R = ( ZZ>= ` C ) )
65	64	eleq2d 2304	. . . . 5 |- ( ph -> ( v e. dom ran R <-> v e. ( ZZ>= ` C ) ) )
66	65	pm5.32i 454	. . . 4 |- ( ( ph /\ v e. dom ran R ) <-> ( ph /\ v e. ( ZZ>= ` C ) ) )
67		df-br 4112	. . . . 5 |- ( v ran R z <-> <. v , z >. e. ran R )
68	67	mobii 2119	. . . 4 |- ( E* z v ran R z <-> E* z <. v , z >. e. ran R )
69	63, 66, 68	3imtr4i 201	. . 3 |- ( ( ph /\ v e. dom ran R ) -> E* z v ran R z )
70	69	ralrimiva 2617	. 2 |- ( ph -> A. v e. dom ran R E* z v ran R z )
71		dffun7 5381	. 2 |- ( Fun ran R <-> ( Rel ran R /\ A. v e. dom ran R E* z v ran R z ) )
72	12, 70, 71	sylanbrc 417	1 |- ( ph -> Fun ran R )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1396   = wceq 1398   E.wex 1541   E*wmo 2083   e. wcel 2205   A.wral 2522   E.wrex 2523   _Vcvv 2815   C_ wss 3213   <.cop 3694   class class class wbr 4111   |-> cmpt 4173   omcom 4714   X. cxp 4749   `'ccnv 4750   dom cdm 4751   ran crn 4752   Rel wrel 4756   Fun wfun 5348   Fn wfn 5349   -->wf 5350   -1-1-onto->wf1o 5353   ` cfv 5354  (class class class)co 6052   e. cmpo 6054   1stc1st 6334   2ndc2nd 6335  freccfrec 6623   1c1 8130   + caddc 8132   ZZcz 9579   ZZ>=cuz 9856

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857

This theorem is referenced by:  frecuzrdgfun  10786