Theorem seqomlem2 7128

Description: Lemma for seq_𝜔. (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
seqomlem2	|- ( Q " om ) Fn om

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

Allowed substitution hints:   I( v, i)

Proof of Theorem seqomlem2

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

Step	Hyp	Ref	Expression
1		frfnom 7112	. . . . . . 7 |- ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) Fn om
2		seqomlem.a	. . . . . . . . 9 |- Q = rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. )
3	2	reseq1i 5275	. . . . . . . 8 |- ( Q |` om ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om )
4	3	fneq1i 5681	. . . . . . 7 |- ( ( Q |` om ) Fn om <-> ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) Fn om )
5	1, 4	mpbir 209	. . . . . 6 |- ( Q |` om ) Fn om
6		fvres 5886	. . . . . . . . 9 |- ( b e. om -> ( ( Q |` om ) ` b ) = ( Q ` b ) )
7	2	seqomlem1 7127	. . . . . . . . 9 |- ( b e. om -> ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. )
8	6, 7	eqtrd 2508	. . . . . . . 8 |- ( b e. om -> ( ( Q |` om ) ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. )
9		fvex 5882	. . . . . . . . 9 |- ( 2nd ` ( Q ` b ) ) e. _V
10		opelxpi 5037	. . . . . . . . 9 |- ( ( b e. om /\ ( 2nd ` ( Q ` b ) ) e. _V ) -> <. b , ( 2nd ` ( Q ` b ) ) >. e. ( om X. _V ) )
11	9, 10	mpan2 671	. . . . . . . 8 |- ( b e. om -> <. b , ( 2nd ` ( Q ` b ) ) >. e. ( om X. _V ) )
12	8, 11	eqeltrd 2555	. . . . . . 7 |- ( b e. om -> ( ( Q |` om ) ` b ) e. ( om X. _V ) )
13	12	rgen 2827	. . . . . 6 |- A. b e. om ( ( Q |` om ) ` b ) e. ( om X. _V )
14		ffnfv 6058	. . . . . 6 |- ( ( Q |` om ) : om --> ( om X. _V ) <-> ( ( Q |` om ) Fn om /\ A. b e. om ( ( Q |` om ) ` b ) e. ( om X. _V ) ) )
15	5, 13, 14	mpbir2an 918	. . . . 5 |- ( Q |` om ) : om --> ( om X. _V )
16		frn 5743	. . . . 5 |- ( ( Q |` om ) : om --> ( om X. _V ) -> ran ( Q |` om ) C_ ( om X. _V ) )
17	15, 16	ax-mp 5	. . . 4 |- ran ( Q |` om ) C_ ( om X. _V )
18		df-br 4454	. . . . . . . . . 10 |- ( a ran ( Q |` om ) b <-> <. a , b >. e. ran ( Q |` om ) )
19		fvelrnb 5921	. . . . . . . . . . 11 |- ( ( Q |` om ) Fn om -> ( <. a , b >. e. ran ( Q |` om ) <-> E. c e. om ( ( Q |` om ) ` c ) = <. a , b >. ) )
20	5, 19	ax-mp 5	. . . . . . . . . 10 |- ( <. a , b >. e. ran ( Q |` om ) <-> E. c e. om ( ( Q |` om ) ` c ) = <. a , b >. )
21		fvres 5886	. . . . . . . . . . . 12 |- ( c e. om -> ( ( Q |` om ) ` c ) = ( Q ` c ) )
22	21	eqeq1d 2469	. . . . . . . . . . 11 |- ( c e. om -> ( ( ( Q |` om ) ` c ) = <. a , b >. <-> ( Q ` c ) = <. a , b >. ) )
23	22	rexbiia 2968	. . . . . . . . . 10 |- ( E. c e. om ( ( Q |` om ) ` c ) = <. a , b >. <-> E. c e. om ( Q ` c ) = <. a , b >. )
24	18, 20, 23	3bitri 271	. . . . . . . . 9 |- ( a ran ( Q |` om ) b <-> E. c e. om ( Q ` c ) = <. a , b >. )
25	2	seqomlem1 7127	. . . . . . . . . . . . . . . 16 |- ( c e. om -> ( Q ` c ) = <. c , ( 2nd ` ( Q ` c ) ) >. )
26	25	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( a e. om /\ c e. om ) -> ( Q ` c ) = <. c , ( 2nd ` ( Q ` c ) ) >. )
27	26	eqeq1d 2469	. . . . . . . . . . . . . 14 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. <-> <. c , ( 2nd ` ( Q ` c ) ) >. = <. a , b >. ) )
28		vex 3121	. . . . . . . . . . . . . . 15 |- c e. _V
29		fvex 5882	. . . . . . . . . . . . . . 15 |- ( 2nd ` ( Q ` c ) ) e. _V
30	28, 29	opth1 4726	. . . . . . . . . . . . . 14 |- ( <. c , ( 2nd ` ( Q ` c ) ) >. = <. a , b >. -> c = a )
31	27, 30	syl6bi 228	. . . . . . . . . . . . 13 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. -> c = a ) )
32		fveq2 5872	. . . . . . . . . . . . . . 15 |- ( c = a -> ( Q ` c ) = ( Q ` a ) )
33	32	eqeq1d 2469	. . . . . . . . . . . . . 14 |- ( c = a -> ( ( Q ` c ) = <. a , b >. <-> ( Q ` a ) = <. a , b >. ) )
34	33	biimpd 207	. . . . . . . . . . . . 13 |- ( c = a -> ( ( Q ` c ) = <. a , b >. -> ( Q ` a ) = <. a , b >. ) )
35	31, 34	syli 37	. . . . . . . . . . . 12 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. -> ( Q ` a ) = <. a , b >. ) )
36		fveq2 5872	. . . . . . . . . . . . 13 |- ( ( Q ` a ) = <. a , b >. -> ( 2nd ` ( Q ` a ) ) = ( 2nd ` <. a , b >. ) )
37		vex 3121	. . . . . . . . . . . . . 14 |- a e. _V
38		vex 3121	. . . . . . . . . . . . . 14 |- b e. _V
39	37, 38	op2nd 6804	. . . . . . . . . . . . 13 |- ( 2nd ` <. a , b >. ) = b
40	36, 39	syl6req 2525	. . . . . . . . . . . 12 |- ( ( Q ` a ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) )
41	35, 40	syl6 33	. . . . . . . . . . 11 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) ) )
42	41	rexlimdva 2959	. . . . . . . . . 10 |- ( a e. om -> ( E. c e. om ( Q ` c ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) ) )
43	2	seqomlem1 7127	. . . . . . . . . . . 12 |- ( a e. om -> ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
44	32	eqeq1d 2469	. . . . . . . . . . . . 13 |- ( c = a -> ( ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. <-> ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) )
45	44	rspcev 3219	. . . . . . . . . . . 12 |- ( ( a e. om /\ ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) -> E. c e. om ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
46	43, 45	mpdan 668	. . . . . . . . . . 11 |- ( a e. om -> E. c e. om ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
47		opeq2 4220	. . . . . . . . . . . . 13 |- ( b = ( 2nd ` ( Q ` a ) ) -> <. a , b >. = <. a , ( 2nd ` ( Q ` a ) ) >. )
48	47	eqeq2d 2481	. . . . . . . . . . . 12 |- ( b = ( 2nd ` ( Q ` a ) ) -> ( ( Q ` c ) = <. a , b >. <-> ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) )
49	48	rexbidv 2978	. . . . . . . . . . 11 |- ( b = ( 2nd ` ( Q ` a ) ) -> ( E. c e. om ( Q ` c ) = <. a , b >. <-> E. c e. om ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) )
50	46, 49	syl5ibrcom 222	. . . . . . . . . 10 |- ( a e. om -> ( b = ( 2nd ` ( Q ` a ) ) -> E. c e. om ( Q ` c ) = <. a , b >. ) )
51	42, 50	impbid 191	. . . . . . . . 9 |- ( a e. om -> ( E. c e. om ( Q ` c ) = <. a , b >. <-> b = ( 2nd ` ( Q ` a ) ) ) )
52	24, 51	syl5bb 257	. . . . . . . 8 |- ( a e. om -> ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) )
53	52	alrimiv 1695	. . . . . . 7 |- ( a e. om -> A. b ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) )
54		fvex 5882	. . . . . . . 8 |- ( 2nd ` ( Q ` a ) ) e. _V
55		eqeq2 2482	. . . . . . . . . 10 |- ( c = ( 2nd ` ( Q ` a ) ) -> ( b = c <-> b = ( 2nd ` ( Q ` a ) ) ) )
56	55	bibi2d 318	. . . . . . . . 9 |- ( c = ( 2nd ` ( Q ` a ) ) -> ( ( a ran ( Q |` om ) b <-> b = c ) <-> ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) ) )
57	56	albidv 1689	. . . . . . . 8 |- ( c = ( 2nd ` ( Q ` a ) ) -> ( A. b ( a ran ( Q |` om ) b <-> b = c ) <-> A. b ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) ) )
58	54, 57	spcev 3210	. . . . . . 7 |- ( A. b ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) -> E. c A. b ( a ran ( Q |` om ) b <-> b = c ) )
59	53, 58	syl 16	. . . . . 6 |- ( a e. om -> E. c A. b ( a ran ( Q |` om ) b <-> b = c ) )
60		df-eu 2279	. . . . . 6 |- ( E! b a ran ( Q |` om ) b <-> E. c A. b ( a ran ( Q |` om ) b <-> b = c ) )
61	59, 60	sylibr 212	. . . . 5 |- ( a e. om -> E! b a ran ( Q |` om ) b )
62	61	rgen 2827	. . . 4 |- A. a e. om E! b a ran ( Q |` om ) b
63		dff3 6045	. . . 4 |- ( ran ( Q |` om ) : om --> _V <-> ( ran ( Q |` om ) C_ ( om X. _V ) /\ A. a e. om E! b a ran ( Q |` om ) b ) )
64	17, 62, 63	mpbir2an 918	. . 3 |- ran ( Q |` om ) : om --> _V
65		df-ima 5018	. . . 4 |- ( Q " om ) = ran ( Q |` om )
66	65	feq1i 5729	. . 3 |- ( ( Q " om ) : om --> _V <-> ran ( Q |` om ) : om --> _V )
67	64, 66	mpbir 209	. 2 |- ( Q " om ) : om --> _V
68		dffn2 5738	. 2 |- ( ( Q " om ) Fn om <-> ( Q " om ) : om --> _V )
69	67, 68	mpbir 209	1 |- ( Q " om ) Fn om

Syntax hints:   <-> wb 184   /\ wa 369   A.wal 1377   = wceq 1379   E.wex 1596   e. wcel 1767   E!weu 2275   A.wral 2817   E.wrex 2818   _Vcvv 3118   C_ wss 3481   (/)c0 3790   <.cop 4039   class class class wbr 4453   _I cid 4796   suc csuc 4886   X. cxp 5003   ran crn 5006   |` cres 5007   "cima 5008   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   |-> cmpt2 6297   omcom 6695   2ndc2nd 6794   reccrdg 7087

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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088

This theorem is referenced by:  seqomlem3  7129  seqomlem4  7130  fnseqom  7132