Theorem frecuzrdgfunlem 10313

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 10309	. . . . 5 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
7		frn 5327	. . . . 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 4693	. . . 4 |- ( ( ZZ>= ` C ) X. S ) C_ ( _V X. _V )
10	8, 9	sstrdi 3140	. . 3 |- ( ph -> ran R C_ ( _V X. _V ) )
11		df-rel 4592	. . 3 |- ( Rel ran R <-> ran R C_ ( _V X. _V ) )
12	10, 11	sylibr 133	. 2 |- ( ph -> Rel ran R )
13		frecuzrdgfunlem.g	. . . . . . . . . 10 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
14	1, 13	frec2uzf1od 10300	. . . . . . . . 9 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
15		f1ocnvdm 5728	. . . . . . . . 9 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
16	14, 15	sylan 281	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
17	6	ffvelrnda 5601	. . . . . . . 8 |- ( ( ph /\ ( `' G ` v ) e. om ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
18	16, 17	syldan 280	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
19		xp2nd 6111	. . . . . . 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 5318	. . . . . . . . . 10 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> R Fn om )
22		fvelrnb 5515	. . . . . . . . . 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	ffvelrnda 5601	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> ( R ` w ) e. ( ( ZZ>= ` C ) X. S ) )
25		1st2nd2 6120	. . . . . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> C e. ZZ )
28	2	adantr 274	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> A e. S )
29	3	adantr 274	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> S C_ T )
30	4	adantlr 469	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ w e. om ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
31		simpr 109	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w e. om ) -> w e. om )
32	27, 28, 29, 30, 5, 31, 13	frecuzrdgg 10310	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w e. om ) -> ( 1st ` ( R ` w ) ) = ( G ` w ) )
33	32	opeq1d 3747	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w e. om ) -> <. ( 1st ` ( R ` w ) ) , ( 2nd ` ( R ` w ) ) >. = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
34	26, 33	eqtrd 2190	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. om ) -> ( R ` w ) = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
35	34	eqeq1d 2166	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. <-> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. ) )
36		vex 2715	. . . . . . . . . . . . . . . . . 18 |- v e. _V
37		vex 2715	. . . . . . . . . . . . . . . . . 18 |- z e. _V
38	36, 37	opth2 4200	. . . . . . . . . . . . . . . . 17 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. <-> ( ( G ` w ) = v /\ ( 2nd ` ( R ` w ) ) = z ) )
39	38	simplbi 272	. . . . . . . . . . . . . . . 16 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. -> ( G ` w ) = v )
40	35, 39	syl6bi 162	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> ( G ` w ) = v ) )
41		f1ocnvfv 5726	. . . . . . . . . . . . . . . 16 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
42	14, 41	sylan 281	. . . . . . . . . . . . . . 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 5467	. . . . . . . . . . . . . . 15 |- ( ( `' G ` v ) = w -> ( R ` ( `' G ` v ) ) = ( R ` w ) )
45	44	fveq2d 5471	. . . . . . . . . . . . . 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 123	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
48	36, 37	op2ndd 6094	. . . . . . . . . . . . 13 |- ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` w ) ) = z )
49	48	adantl 275	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` w ) ) = z )
50	47, 49	eqtr2d 2191	. . . . . . . . . . 11 |- ( ( ( ph /\ w e. om ) /\ ( R ` w ) = <. v , z >. ) -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) )
51	50	ex 114	. . . . . . . . . 10 |- ( ( ph /\ w e. om ) -> ( ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
52	51	rexlimdva 2574	. . . . . . . . 9 |- ( ph -> ( E. w e. om ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
53	23, 52	sylbid 149	. . . . . . . 8 |- ( ph -> ( <. v , z >. e. ran R -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
54	53	alrimiv 1854	. . . . . . 7 |- ( ph -> A. z ( <. v , z >. e. ran R -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
55	54	adantr 274	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A. z ( <. v , z >. e. ran R -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
56		eqeq2 2167	. . . . . . . . 9 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( z = w <-> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
57	56	imbi2d 229	. . . . . . . 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 1804	. . . . . . 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 2800	. . . . . 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 1508	. . . . . 6 |- F/ w <. v , z >. e. ran R
62	61	mo2r 2058	. . . . 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 10312	. . . . . 6 |- ( ph -> dom ran R = ( ZZ>= ` C ) )
65	64	eleq2d 2227	. . . . 5 |- ( ph -> ( v e. dom ran R <-> v e. ( ZZ>= ` C ) ) )
66	65	pm5.32i 450	. . . 4 |- ( ( ph /\ v e. dom ran R ) <-> ( ph /\ v e. ( ZZ>= ` C ) ) )
67		df-br 3966	. . . . 5 |- ( v ran R z <-> <. v , z >. e. ran R )
68	67	mobii 2043	. . . 4 |- ( E* z v ran R z <-> E* z <. v , z >. e. ran R )
69	63, 66, 68	3imtr4i 200	. . 3 |- ( ( ph /\ v e. dom ran R ) -> E* z v ran R z )
70	69	ralrimiva 2530	. 2 |- ( ph -> A. v e. dom ran R E* z v ran R z )
71		dffun7 5196	. 2 |- ( Fun ran R <-> ( Rel ran R /\ A. v e. dom ran R E* z v ran R z ) )
72	12, 70, 71	sylanbrc 414	1 |- ( ph -> Fun ran R )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   = wceq 1335   E.wex 1472   E*wmo 2007   e. wcel 2128   A.wral 2435   E.wrex 2436   _Vcvv 2712   C_ wss 3102   <.cop 3563   class class class wbr 3965   |-> cmpt 4025   omcom 4548   X. cxp 4583   `'ccnv 4584   dom cdm 4585   ran crn 4586   Rel wrel 4590   Fun wfun 5163   Fn wfn 5164   -->wf 5165   -1-1-onto->wf1o 5168   ` cfv 5169  (class class class)co 5821   e. cmpo 5823   1stc1st 6083   2ndc2nd 6084  freccfrec 6334   1c1 7728   + caddc 7730   ZZcz 9162   ZZ>=cuz 9434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7818  ax-resscn 7819  ax-1cn 7820  ax-1re 7821  ax-icn 7822  ax-addcl 7823  ax-addrcl 7824  ax-mulcl 7825  ax-addcom 7827  ax-addass 7829  ax-distr 7831  ax-i2m1 7832  ax-0lt1 7833  ax-0id 7835  ax-rnegex 7836  ax-cnre 7838  ax-pre-ltirr 7839  ax-pre-ltwlin 7840  ax-pre-lttrn 7841  ax-pre-ltadd 7843

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-frec 6335  df-pnf 7909  df-mnf 7910  df-xr 7911  df-ltxr 7912  df-le 7913  df-sub 8043  df-neg 8044  df-inn 8829  df-n0 9086  df-z 9163  df-uz 9435

This theorem is referenced by:  frecuzrdgfun  10314