Theorem frecuzrdgfunlem 10405

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 10401	. . . . 5 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
7		frn 5370	. . . . 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 4731	. . . 4 |- ( ( ZZ>= ` C ) X. S ) C_ ( _V X. _V )
10	8, 9	sstrdi 3167	. . 3 |- ( ph -> ran R C_ ( _V X. _V ) )
11		df-rel 4630	. . 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 10392	. . . . . . . . 9 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
15		f1ocnvdm 5776	. . . . . . . . 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 5647	. . . . . . . 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 6161	. . . . . . 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 5361	. . . . . . . . . 10 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> R Fn om )
22		fvelrnb 5559	. . . . . . . . . 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 5647	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) )
25		1st2nd2 6170	. . . . . . . . . . . . . . . . . . 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 10402	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> ( 1st ` ( R ` w ) ) = ( G ` w ) )
33	32	opeq1d 3782	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w e. om ) -> <. ( 1st ` ( R ` w ) ) , ( 2nd ` ( R ` w ) ) >. = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
34	26, 33	eqtrd 2210	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. om ) -> ( R ` w ) = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
35	34	eqeq1d 2186	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. <-> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. ) )
36		vex 2740	. . . . . . . . . . . . . . . . . 18 |- v e. _V
37		vex 2740	. . . . . . . . . . . . . . . . . 18 |- z e. _V
38	36, 37	opth2 4237	. . . . . . . . . . . . . . . . 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	syl6bi 163	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( G ` w ) = v ) )
41		f1ocnvfv 5774	. . . . . . . . . . . . . . . 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 5511	. . . . . . . . . . . . . . 15 |- ( ( `' G ` v ) = w -> ( R ` ( `' G ` v ) ) = ( R ` w ) )
45	44	fveq2d 5515	. . . . . . . . . . . . . 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 6144	. . . . . . . . . . . . 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 2211	. . . . . . . . . . 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 2594	. . . . . . . . 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 1874	. . . . . . 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 2187	. . . . . . . . 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 1824	. . . . . . 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 2825	. . . . . 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 1528	. . . . . 6 |- F/ w <. v , z >. e. ran R
62	61	mo2r 2078	. . . . 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 10404	. . . . . 6 |- ( ph -> dom ran R = ( ZZ>= ` C ) )
65	64	eleq2d 2247	. . . . 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 4001	. . . . 5 |- ( v ran R z <-> <. v , z >. e. ran R )
68	67	mobii 2063	. . . 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 2550	. 2 |- ( ph -> A. v e. dom ran R E* z v ran R z )
71		dffun7 5239	. 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 1351   = wceq 1353   E.wex 1492   E*wmo 2027   e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2737   C_ wss 3129   <.cop 3594   class class class wbr 4000   |-> cmpt 4061   omcom 4586   X. cxp 4621   `'ccnv 4622   dom cdm 4623   ran crn 4624   Rel wrel 4628   Fun wfun 5206   Fn wfn 5207   -->wf 5208   -1-1-onto->wf1o 5211   ` cfv 5212  (class class class)co 5869   e. cmpo 5871   1stc1st 6133   2ndc2nd 6134  freccfrec 6385   1c1 7803   + caddc 7805   ZZcz 9242   ZZ>=cuz 9517

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518

This theorem is referenced by:  frecuzrdgfun  10406