Theorem uzrdgfni 12757

Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 12755. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.)

Hypotheses

Ref	Expression
om2uz.1	|- C e. ZZ
om2uz.2	|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )
uzrdg.1	|- A e. _V
uzrdg.2	|- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om )
uzrdg.3	|- S = ran R

Assertion

Ref	Expression
uzrdgfni	|- S Fn ( ZZ>= ` C )

Distinct variable groups:   y, A   x, y, C   y, G   x, F, y

Allowed substitution hints:   A( x)   R( x, y)   S( x, y)   G( x)

Proof of Theorem uzrdgfni

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

Step	Hyp	Ref	Expression
1		uzrdg.3	. . . . . . . . 9 |- S = ran R
2	1	eleq2i 2693	. . . . . . . 8 |- ( z e. S <-> z e. ran R )
3		frfnom 7530	. . . . . . . . . 10 |- ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) Fn om
4		uzrdg.2	. . . . . . . . . . 11 |- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om )
5	4	fneq1i 5985	. . . . . . . . . 10 |- ( R Fn om <-> ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) Fn om )
6	3, 5	mpbir 221	. . . . . . . . 9 |- R Fn om
7		fvelrnb 6243	. . . . . . . . 9 |- ( R Fn om -> ( z e. ran R <-> E. w e. om ( R ` w ) = z ) )
8	6, 7	ax-mp 5	. . . . . . . 8 |- ( z e. ran R <-> E. w e. om ( R ` w ) = z )
9	2, 8	bitri 264	. . . . . . 7 |- ( z e. S <-> E. w e. om ( R ` w ) = z )
10		om2uz.1	. . . . . . . . . . 11 |- C e. ZZ
11		om2uz.2	. . . . . . . . . . 11 |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )
12		uzrdg.1	. . . . . . . . . . 11 |- A e. _V
13	10, 11, 12, 4	om2uzrdg 12755	. . . . . . . . . 10 |- ( w e. om -> ( R ` w ) = <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. )
14	10, 11	om2uzuzi 12748	. . . . . . . . . . 11 |- ( w e. om -> ( G ` w ) e. ( ZZ>= ` C ) )
15		fvex 6201	. . . . . . . . . . 11 |- ( 2nd ` ( R ` w ) ) e. _V
16		opelxpi 5148	. . . . . . . . . . 11 |- ( ( ( G ` w ) e. ( ZZ>= ` C ) /\ ( 2nd ` ( R ` w ) ) e. _V ) -> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. e. ( ( ZZ>= ` C ) X. _V ) )
17	14, 15, 16	sylancl 694	. . . . . . . . . 10 |- ( w e. om -> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. e. ( ( ZZ>= ` C ) X. _V ) )
18	13, 17	eqeltrd 2701	. . . . . . . . 9 |- ( w e. om -> ( R ` w ) e. ( ( ZZ>= ` C ) X. _V ) )
19		eleq1 2689	. . . . . . . . 9 |- ( ( R ` w ) = z -> ( ( R ` w ) e. ( ( ZZ>= ` C ) X. _V ) <-> z e. ( ( ZZ>= ` C ) X. _V ) ) )
20	18, 19	syl5ibcom 235	. . . . . . . 8 |- ( w e. om -> ( ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. _V ) ) )
21	20	rexlimiv 3027	. . . . . . 7 |- ( E. w e. om ( R ` w ) = z -> z e. ( ( ZZ>= ` C ) X. _V ) )
22	9, 21	sylbi 207	. . . . . 6 |- ( z e. S -> z e. ( ( ZZ>= ` C ) X. _V ) )
23	22	ssriv 3607	. . . . 5 |- S C_ ( ( ZZ>= ` C ) X. _V )
24		xpss 5226	. . . . 5 |- ( ( ZZ>= ` C ) X. _V ) C_ ( _V X. _V )
25	23, 24	sstri 3612	. . . 4 |- S C_ ( _V X. _V )
26		df-rel 5121	. . . 4 |- ( Rel S <-> S C_ ( _V X. _V ) )
27	25, 26	mpbir 221	. . 3 |- Rel S
28		fvex 6201	. . . . . 6 |- ( 2nd ` ( R ` ( `' G ` v ) ) ) e. _V
29		eqeq2 2633	. . . . . . . 8 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( z = w <-> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) )
30	29	imbi2d 330	. . . . . . 7 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( ( <. v , z >. e. S -> z = w ) <-> ( <. v , z >. e. S -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
31	30	albidv 1849	. . . . . 6 |- ( w = ( 2nd ` ( R ` ( `' G ` v ) ) ) -> ( A. z ( <. v , z >. e. S -> z = w ) <-> A. z ( <. v , z >. e. S -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) ) )
32	28, 31	spcev 3300	. . . . 5 |- ( A. z ( <. v , z >. e. S -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) ) -> E. w A. z ( <. v , z >. e. S -> z = w ) )
33	1	eleq2i 2693	. . . . . . 7 |- ( <. v , z >. e. S <-> <. v , z >. e. ran R )
34		fvelrnb 6243	. . . . . . . 8 |- ( R Fn om -> ( <. v , z >. e. ran R <-> E. w e. om ( R ` w ) = <. v , z >. ) )
35	6, 34	ax-mp 5	. . . . . . 7 |- ( <. v , z >. e. ran R <-> E. w e. om ( R ` w ) = <. v , z >. )
36	33, 35	bitri 264	. . . . . 6 |- ( <. v , z >. e. S <-> E. w e. om ( R ` w ) = <. v , z >. )
37	13	eqeq1d 2624	. . . . . . . . . . . 12 |- ( w e. om -> ( ( R ` w ) = <. v , z >. <-> <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. ) )
38		fvex 6201	. . . . . . . . . . . . 13 |- ( G ` w ) e. _V
39	38, 15	opth1 4944	. . . . . . . . . . . 12 |- ( <. ( G ` w ) , ( 2nd ` ( R ` w ) ) >. = <. v , z >. -> ( G ` w ) = v )
40	37, 39	syl6bi 243	. . . . . . . . . . 11 |- ( w e. om -> ( ( R ` w ) = <. v , z >. -> ( G ` w ) = v ) )
41	10, 11	om2uzf1oi 12752	. . . . . . . . . . . 12 |- G : om -1-1-onto-> ( ZZ>= ` C )
42		f1ocnvfv 6534	. . . . . . . . . . . 12 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ w e. om ) -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
43	41, 42	mpan 706	. . . . . . . . . . 11 |- ( w e. om -> ( ( G ` w ) = v -> ( `' G ` v ) = w ) )
44	40, 43	syld 47	. . . . . . . . . 10 |- ( w e. om -> ( ( R ` w ) = <. v , z >. -> ( `' G ` v ) = w ) )
45		fveq2 6191	. . . . . . . . . . 11 |- ( ( `' G ` v ) = w -> ( R ` ( `' G ` v ) ) = ( R ` w ) )
46	45	fveq2d 6195	. . . . . . . . . 10 |- ( ( `' G ` v ) = w -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
47	44, 46	syl6 35	. . . . . . . . 9 |- ( w e. om -> ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) ) )
48	47	imp 445	. . . . . . . 8 |- ( ( w e. om /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) = ( 2nd ` ( R ` w ) ) )
49		vex 3203	. . . . . . . . . 10 |- v e. _V
50		vex 3203	. . . . . . . . . 10 |- z e. _V
51	49, 50	op2ndd 7179	. . . . . . . . 9 |- ( ( R ` w ) = <. v , z >. -> ( 2nd ` ( R ` w ) ) = z )
52	51	adantl 482	. . . . . . . 8 |- ( ( w e. om /\ ( R ` w ) = <. v , z >. ) -> ( 2nd ` ( R ` w ) ) = z )
53	48, 52	eqtr2d 2657	. . . . . . 7 |- ( ( w e. om /\ ( R ` w ) = <. v , z >. ) -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) )
54	53	rexlimiva 3028	. . . . . 6 |- ( E. w e. om ( R ` w ) = <. v , z >. -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) )
55	36, 54	sylbi 207	. . . . 5 |- ( <. v , z >. e. S -> z = ( 2nd ` ( R ` ( `' G ` v ) ) ) )
56	32, 55	mpg 1724	. . . 4 |- E. w A. z ( <. v , z >. e. S -> z = w )
57	56	ax-gen 1722	. . 3 |- A. v E. w A. z ( <. v , z >. e. S -> z = w )
58		dffun5 5901	. . 3 |- ( Fun S <-> ( Rel S /\ A. v E. w A. z ( <. v , z >. e. S -> z = w ) ) )
59	27, 57, 58	mpbir2an 955	. 2 |- Fun S
60		dmss 5323	. . . . 5 |- ( S C_ ( ( ZZ>= ` C ) X. _V ) -> dom S C_ dom ( ( ZZ>= ` C ) X. _V ) )
61	23, 60	ax-mp 5	. . . 4 |- dom S C_ dom ( ( ZZ>= ` C ) X. _V )
62		dmxpss 5565	. . . 4 |- dom ( ( ZZ>= ` C ) X. _V ) C_ ( ZZ>= ` C )
63	61, 62	sstri 3612	. . 3 |- dom S C_ ( ZZ>= ` C )
64	10, 11, 12, 4	uzrdglem 12756	. . . . . 6 |- ( v e. ( ZZ>= ` C ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R )
65	64, 1	syl6eleqr 2712	. . . . 5 |- ( v e. ( ZZ>= ` C ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. S )
66	49, 28	opeldm 5328	. . . . 5 |- ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. S -> v e. dom S )
67	65, 66	syl 17	. . . 4 |- ( v e. ( ZZ>= ` C ) -> v e. dom S )
68	67	ssriv 3607	. . 3 |- ( ZZ>= ` C ) C_ dom S
69	63, 68	eqssi 3619	. 2 |- dom S = ( ZZ>= ` C )
70		df-fn 5891	. 2 |- ( S Fn ( ZZ>= ` C ) <-> ( Fun S /\ dom S = ( ZZ>= ` C ) ) )
71	59, 69, 70	mpbir2an 955	1 |- S Fn ( ZZ>= ` C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   <.cop 4183   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   2ndc2nd 7167   reccrdg 7505   1c1 9937   + caddc 9939   ZZcz 11377   ZZ>=cuz 11687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688

This theorem is referenced by:  uzrdg0i  12758  uzrdgsuci  12759  seqfn  12813