Theorem seqomlem2 6906

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 6890	. . . . . . 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 5106	. . . . . . . 8 |- ( Q |` om ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om )
4	3	fneq1i 5505	. . . . . . 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 5704	. . . . . . . . 9 |- ( b e. om -> ( ( Q |` om ) ` b ) = ( Q ` b ) )
7	2	seqomlem1 6905	. . . . . . . . 9 |- ( b e. om -> ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. )
8	6, 7	eqtrd 2475	. . . . . . . 8 |- ( b e. om -> ( ( Q |` om ) ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. )
9		fvex 5701	. . . . . . . . 9 |- ( 2nd ` ( Q ` b ) ) e. _V
10		opelxpi 4871	. . . . . . . . 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 2517	. . . . . . 7 |- ( b e. om -> ( ( Q |` om ) ` b ) e. ( om X. _V ) )
13	12	rgen 2781	. . . . . 6 |- A. b e. om ( ( Q |` om ) ` b ) e. ( om X. _V )
14		ffnfv 5869	. . . . . 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 911	. . . . 5 |- ( Q |` om ) : om --> ( om X. _V )
16		frn 5565	. . . . 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 4293	. . . . . . . . . 10 |- ( a ran ( Q |` om ) b <-> <. a , b >. e. ran ( Q |` om ) )
19		fvelrnb 5739	. . . . . . . . . . 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 5704	. . . . . . . . . . . 12 |- ( c e. om -> ( ( Q |` om ) ` c ) = ( Q ` c ) )
22	21	eqeq1d 2451	. . . . . . . . . . 11 |- ( c e. om -> ( ( ( Q |` om ) ` c ) = <. a , b >. <-> ( Q ` c ) = <. a , b >. ) )
23	22	rexbiia 2748	. . . . . . . . . 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 6905	. . . . . . . . . . . . . . . 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 2451	. . . . . . . . . . . . . 14 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. <-> <. c , ( 2nd ` ( Q ` c ) ) >. = <. a , b >. ) )
28		vex 2975	. . . . . . . . . . . . . . 15 |- c e. _V
29		fvex 5701	. . . . . . . . . . . . . . 15 |- ( 2nd ` ( Q ` c ) ) e. _V
30	28, 29	opth1 4565	. . . . . . . . . . . . . 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 5691	. . . . . . . . . . . . . . 15 |- ( c = a -> ( Q ` c ) = ( Q ` a ) )
33	32	eqeq1d 2451	. . . . . . . . . . . . . 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 5691	. . . . . . . . . . . . 13 |- ( ( Q ` a ) = <. a , b >. -> ( 2nd ` ( Q ` a ) ) = ( 2nd ` <. a , b >. ) )
37		vex 2975	. . . . . . . . . . . . . 14 |- a e. _V
38		vex 2975	. . . . . . . . . . . . . 14 |- b e. _V
39	37, 38	op2nd 6586	. . . . . . . . . . . . 13 |- ( 2nd ` <. a , b >. ) = b
40	36, 39	syl6req 2492	. . . . . . . . . . . 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 2841	. . . . . . . . . 10 |- ( a e. om -> ( E. c e. om ( Q ` c ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) ) )
43	2	seqomlem1 6905	. . . . . . . . . . . 12 |- ( a e. om -> ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
44	32	eqeq1d 2451	. . . . . . . . . . . . 13 |- ( c = a -> ( ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. <-> ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) )
45	44	rspcev 3073	. . . . . . . . . . . 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 4060	. . . . . . . . . . . . 13 |- ( b = ( 2nd ` ( Q ` a ) ) -> <. a , b >. = <. a , ( 2nd ` ( Q ` a ) ) >. )
48	47	eqeq2d 2454	. . . . . . . . . . . 12 |- ( b = ( 2nd ` ( Q ` a ) ) -> ( ( Q ` c ) = <. a , b >. <-> ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) )
49	48	rexbidv 2736	. . . . . . . . . . 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 1685	. . . . . . 7 |- ( a e. om -> A. b ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) )
54		fvex 5701	. . . . . . . 8 |- ( 2nd ` ( Q ` a ) ) e. _V
55		eqeq2 2452	. . . . . . . . . 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 1679	. . . . . . . 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 3064	. . . . . . 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 2257	. . . . . 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 2781	. . . 4 |- A. a e. om E! b a ran ( Q |` om ) b
63		dff3 5856	. . . 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 911	. . 3 |- ran ( Q |` om ) : om --> _V
65		df-ima 4853	. . . 4 |- ( Q " om ) = ran ( Q |` om )
66	65	feq1i 5551	. . 3 |- ( ( Q " om ) : om --> _V <-> ran ( Q |` om ) : om --> _V )
67	64, 66	mpbir 209	. 2 |- ( Q " om ) : om --> _V
68		dffn2 5560	. 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 1367   = wceq 1369   E.wex 1586   e. wcel 1756   E!weu 2253   A.wral 2715   E.wrex 2716   _Vcvv 2972   C_ wss 3328   (/)c0 3637   <.cop 3883   class class class wbr 4292   _I cid 4631   suc csuc 4721   X. cxp 4838   ran crn 4841   |` cres 4842   "cima 4843   Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   e. cmpt2 6093   omcom 6476   2ndc2nd 6576   reccrdg 6865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-2nd 6578  df-recs 6832  df-rdg 6866

This theorem is referenced by:  seqomlem3  6907  seqomlem4  6908  fnseqom  6910