Theorem seqomlem1 7112

Description: Lemma for seq_𝜔. The underlying recursion generates a sequence of pairs with the expected first values. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)

Hypothesis

Ref	Expression
seqomlem.a	|- Q = rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. )

Assertion

Ref	Expression
seqomlem1	|- ( A e. om -> ( Q ` A ) = <. A , ( 2nd ` ( Q ` A ) ) >. )

Distinct variable groups:   Q, i, v   A, i, v   i, F, v

Allowed substitution hints:   I( v, i)

Proof of Theorem seqomlem1

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5864	. . 3 |- ( a = (/) -> ( Q ` a ) = ( Q ` (/) ) )
2		id 22	. . . 4 |- ( a = (/) -> a = (/) )
3	1	fveq2d 5868	. . . 4 |- ( a = (/) -> ( 2nd ` ( Q ` a ) ) = ( 2nd ` ( Q ` (/) ) ) )
4	2, 3	opeq12d 4221	. . 3 |- ( a = (/) -> <. a , ( 2nd ` ( Q ` a ) ) >. = <. (/) , ( 2nd ` ( Q ` (/) ) ) >. )
5	1, 4	eqeq12d 2489	. 2 |- ( a = (/) -> ( ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. <-> ( Q ` (/) ) = <. (/) , ( 2nd ` ( Q ` (/) ) ) >. ) )
6		fveq2 5864	. . 3 |- ( a = b -> ( Q ` a ) = ( Q ` b ) )
7		id 22	. . . 4 |- ( a = b -> a = b )
8	6	fveq2d 5868	. . . 4 |- ( a = b -> ( 2nd ` ( Q ` a ) ) = ( 2nd ` ( Q ` b ) ) )
9	7, 8	opeq12d 4221	. . 3 |- ( a = b -> <. a , ( 2nd ` ( Q ` a ) ) >. = <. b , ( 2nd ` ( Q ` b ) ) >. )
10	6, 9	eqeq12d 2489	. 2 |- ( a = b -> ( ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. <-> ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. ) )
11		fveq2 5864	. . 3 |- ( a = suc b -> ( Q ` a ) = ( Q ` suc b ) )
12		id 22	. . . 4 |- ( a = suc b -> a = suc b )
13	11	fveq2d 5868	. . . 4 |- ( a = suc b -> ( 2nd ` ( Q ` a ) ) = ( 2nd ` ( Q ` suc b ) ) )
14	12, 13	opeq12d 4221	. . 3 |- ( a = suc b -> <. a , ( 2nd ` ( Q ` a ) ) >. = <. suc b , ( 2nd ` ( Q ` suc b ) ) >. )
15	11, 14	eqeq12d 2489	. 2 |- ( a = suc b -> ( ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. <-> ( Q ` suc b ) = <. suc b , ( 2nd ` ( Q ` suc b ) ) >. ) )
16		fveq2 5864	. . 3 |- ( a = A -> ( Q ` a ) = ( Q ` A ) )
17		id 22	. . . 4 |- ( a = A -> a = A )
18	16	fveq2d 5868	. . . 4 |- ( a = A -> ( 2nd ` ( Q ` a ) ) = ( 2nd ` ( Q ` A ) ) )
19	17, 18	opeq12d 4221	. . 3 |- ( a = A -> <. a , ( 2nd ` ( Q ` a ) ) >. = <. A , ( 2nd ` ( Q ` A ) ) >. )
20	16, 19	eqeq12d 2489	. 2 |- ( a = A -> ( ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. <-> ( Q ` A ) = <. A , ( 2nd ` ( Q ` A ) ) >. ) )
21		seqomlem.a	. . . . 5 |- Q = rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. )
22	21	fveq1i 5865	. . . 4 |- ( Q ` (/) ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` (/) )
23		opex 4711	. . . . 5 |- <. (/) , ( _I ` I ) >. e. _V
24	23	rdg0 7084	. . . 4 |- ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` (/) ) = <. (/) , ( _I ` I ) >.
25	22, 24	eqtri 2496	. . 3 |- ( Q ` (/) ) = <. (/) , ( _I ` I ) >.
26		0ex 4577	. . . . . . 7 |- (/) e. _V
27		fvex 5874	. . . . . . 7 |- ( _I ` I ) e. _V
28	26, 27	op2nd 6790	. . . . . 6 |- ( 2nd ` <. (/) , ( _I ` I ) >. ) = ( _I ` I )
29	28	eqcomi 2480	. . . . 5 |- ( _I ` I ) = ( 2nd ` <. (/) , ( _I ` I ) >. )
30	29	opeq2i 4217	. . . 4 |- <. (/) , ( _I ` I ) >. = <. (/) , ( 2nd ` <. (/) , ( _I ` I ) >. ) >.
31		id 22	. . . 4 |- ( ( Q ` (/) ) = <. (/) , ( _I ` I ) >. -> ( Q ` (/) ) = <. (/) , ( _I ` I ) >. )
32		fveq2 5864	. . . . 5 |- ( ( Q ` (/) ) = <. (/) , ( _I ` I ) >. -> ( 2nd ` ( Q ` (/) ) ) = ( 2nd ` <. (/) , ( _I ` I ) >. ) )
33	32	opeq2d 4220	. . . 4 |- ( ( Q ` (/) ) = <. (/) , ( _I ` I ) >. -> <. (/) , ( 2nd ` ( Q ` (/) ) ) >. = <. (/) , ( 2nd ` <. (/) , ( _I ` I ) >. ) >. )
34	30, 31, 33	3eqtr4a 2534	. . 3 |- ( ( Q ` (/) ) = <. (/) , ( _I ` I ) >. -> ( Q ` (/) ) = <. (/) , ( 2nd ` ( Q ` (/) ) ) >. )
35	25, 34	ax-mp 5	. 2 |- ( Q ` (/) ) = <. (/) , ( 2nd ` ( Q ` (/) ) ) >.
36		df-ov 6285	. . . . . 6 |- ( b ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` b ) ) ) = ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` <. b , ( 2nd ` ( Q ` b ) ) >. )
37		fvex 5874	. . . . . . 7 |- ( 2nd ` ( Q ` b ) ) e. _V
38		suceq 4943	. . . . . . . . 9 |- ( i = b -> suc i = suc b )
39		oveq1 6289	. . . . . . . . 9 |- ( i = b -> ( i F v ) = ( b F v ) )
40	38, 39	opeq12d 4221	. . . . . . . 8 |- ( i = b -> <. suc i , ( i F v ) >. = <. suc b , ( b F v ) >. )
41		oveq2 6290	. . . . . . . . 9 |- ( v = ( 2nd ` ( Q ` b ) ) -> ( b F v ) = ( b F ( 2nd ` ( Q ` b ) ) ) )
42	41	opeq2d 4220	. . . . . . . 8 |- ( v = ( 2nd ` ( Q ` b ) ) -> <. suc b , ( b F v ) >. = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. )
43		eqid 2467	. . . . . . . 8 |- ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) = ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. )
44		opex 4711	. . . . . . . 8 |- <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. e. _V
45	40, 42, 43, 44	ovmpt2 6420	. . . . . . 7 |- ( ( b e. om /\ ( 2nd ` ( Q ` b ) ) e. _V ) -> ( b ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` b ) ) ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. )
46	37, 45	mpan2 671	. . . . . 6 |- ( b e. om -> ( b ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` b ) ) ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. )
47	36, 46	syl5eqr 2522	. . . . 5 |- ( b e. om -> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` <. b , ( 2nd ` ( Q ` b ) ) >. ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. )
48		fveq2 5864	. . . . . 6 |- ( ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. -> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` <. b , ( 2nd ` ( Q ` b ) ) >. ) )
49	48	eqeq1d 2469	. . . . 5 |- ( ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. -> ( ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. <-> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` <. b , ( 2nd ` ( Q ` b ) ) >. ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. ) )
50	47, 49	syl5ibrcom 222	. . . 4 |- ( b e. om -> ( ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. -> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. ) )
51		vex 3116	. . . . . . . . . 10 |- b e. _V
52	51	sucex 6624	. . . . . . . . 9 |- suc b e. _V
53		ovex 6307	. . . . . . . . 9 |- ( b F ( 2nd ` ( Q ` b ) ) ) e. _V
54	52, 53	op2nd 6790	. . . . . . . 8 |- ( 2nd ` <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. ) = ( b F ( 2nd ` ( Q ` b ) ) )
55	54	eqcomi 2480	. . . . . . 7 |- ( b F ( 2nd ` ( Q ` b ) ) ) = ( 2nd ` <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. )
56	55	a1i 11	. . . . . 6 |- ( b e. om -> ( b F ( 2nd ` ( Q ` b ) ) ) = ( 2nd ` <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. ) )
57	56	opeq2d 4220	. . . . 5 |- ( b e. om -> <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. = <. suc b , ( 2nd ` <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. ) >. )
58		id 22	. . . . . 6 |- ( ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. -> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. )
59		fveq2 5864	. . . . . . 7 |- ( ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. -> ( 2nd ` ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) ) = ( 2nd ` <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. ) )
60	59	opeq2d 4220	. . . . . 6 |- ( ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. -> <. suc b , ( 2nd ` ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) ) >. = <. suc b , ( 2nd ` <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. ) >. )
61	58, 60	eqeq12d 2489	. . . . 5 |- ( ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. -> ( ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( 2nd ` ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) ) >. <-> <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. = <. suc b , ( 2nd ` <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. ) >. ) )
62	57, 61	syl5ibrcom 222	. . . 4 |- ( b e. om -> ( ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( b F ( 2nd ` ( Q ` b ) ) ) >. -> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( 2nd ` ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) ) >. ) )
63	50, 62	syld 44	. . 3 |- ( b e. om -> ( ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. -> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( 2nd ` ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) ) >. ) )
64		frsuc 7099	. . . . 5 |- ( b e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` suc b ) = ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` b ) ) )
65		peano2 6698	. . . . . . 7 |- ( b e. om -> suc b e. om )
66		fvres 5878	. . . . . . 7 |- ( suc b e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` suc b ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` suc b ) )
67	65, 66	syl 16	. . . . . 6 |- ( b e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` suc b ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` suc b ) )
68	21	fveq1i 5865	. . . . . 6 |- ( Q ` suc b ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` suc b )
69	67, 68	syl6eqr 2526	. . . . 5 |- ( b e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` suc b ) = ( Q ` suc b ) )
70		fvres 5878	. . . . . . 7 |- ( b e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` b ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` b ) )
71	21	fveq1i 5865	. . . . . . 7 |- ( Q ` b ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` b )
72	70, 71	syl6eqr 2526	. . . . . 6 |- ( b e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` b ) = ( Q ` b ) )
73	72	fveq2d 5868	. . . . 5 |- ( b e. om -> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` b ) ) = ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) )
74	64, 69, 73	3eqtr3d 2516	. . . 4 |- ( b e. om -> ( Q ` suc b ) = ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) )
75	74	fveq2d 5868	. . . . 5 |- ( b e. om -> ( 2nd ` ( Q ` suc b ) ) = ( 2nd ` ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) ) )
76	75	opeq2d 4220	. . . 4 |- ( b e. om -> <. suc b , ( 2nd ` ( Q ` suc b ) ) >. = <. suc b , ( 2nd ` ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) ) >. )
77	74, 76	eqeq12d 2489	. . 3 |- ( b e. om -> ( ( Q ` suc b ) = <. suc b , ( 2nd ` ( Q ` suc b ) ) >. <-> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) = <. suc b , ( 2nd ` ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` b ) ) ) >. ) )
78	63, 77	sylibrd 234	. 2 |- ( b e. om -> ( ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. -> ( Q ` suc b ) = <. suc b , ( 2nd ` ( Q ` suc b ) ) >. ) )
79	5, 10, 15, 20, 35, 78	finds 6704	1 |- ( A e. om -> ( Q ` A ) = <. A , ( 2nd ` ( Q ` A ) ) >. )

Syntax hints:   -> wi 4   = wceq 1379   e. wcel 1767   _Vcvv 3113   (/)c0 3785   <.cop 4033   _I cid 4790   suc csuc 4880   |` cres 5001   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   omcom 6678   2ndc2nd 6780   reccrdg 7072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073

This theorem is referenced by:  seqomlem2  7113  seqomlem4  7115