Theorem seqomlem2 7546

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 7530	. . . . . . 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 5392	. . . . . . . 8 |- ( Q |` om ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om )
4	3	fneq1i 5985	. . . . . . 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 221	. . . . . 6 |- ( Q |` om ) Fn om
6		fvres 6207	. . . . . . . . 9 |- ( b e. om -> ( ( Q |` om ) ` b ) = ( Q ` b ) )
7	2	seqomlem1 7545	. . . . . . . . 9 |- ( b e. om -> ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. )
8	6, 7	eqtrd 2656	. . . . . . . 8 |- ( b e. om -> ( ( Q |` om ) ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. )
9		fvex 6201	. . . . . . . . 9 |- ( 2nd ` ( Q ` b ) ) e. _V
10		opelxpi 5148	. . . . . . . . 9 |- ( ( b e. om /\ ( 2nd ` ( Q ` b ) ) e. _V ) -> <. b , ( 2nd ` ( Q ` b ) ) >. e. ( om X. _V ) )
11	9, 10	mpan2 707	. . . . . . . 8 |- ( b e. om -> <. b , ( 2nd ` ( Q ` b ) ) >. e. ( om X. _V ) )
12	8, 11	eqeltrd 2701	. . . . . . 7 |- ( b e. om -> ( ( Q |` om ) ` b ) e. ( om X. _V ) )
13	12	rgen 2922	. . . . . 6 |- A. b e. om ( ( Q |` om ) ` b ) e. ( om X. _V )
14		ffnfv 6388	. . . . . 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 955	. . . . 5 |- ( Q |` om ) : om --> ( om X. _V )
16		frn 6053	. . . . 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 4654	. . . . . . . . . 10 |- ( a ran ( Q |` om ) b <-> <. a , b >. e. ran ( Q |` om ) )
19		fvelrnb 6243	. . . . . . . . . . 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 6207	. . . . . . . . . . . 12 |- ( c e. om -> ( ( Q |` om ) ` c ) = ( Q ` c ) )
22	21	eqeq1d 2624	. . . . . . . . . . 11 |- ( c e. om -> ( ( ( Q |` om ) ` c ) = <. a , b >. <-> ( Q ` c ) = <. a , b >. ) )
23	22	rexbiia 3040	. . . . . . . . . 10 |- ( E. c e. om ( ( Q |` om ) ` c ) = <. a , b >. <-> E. c e. om ( Q ` c ) = <. a , b >. )
24	18, 20, 23	3bitri 286	. . . . . . . . 9 |- ( a ran ( Q |` om ) b <-> E. c e. om ( Q ` c ) = <. a , b >. )
25	2	seqomlem1 7545	. . . . . . . . . . . . . . . 16 |- ( c e. om -> ( Q ` c ) = <. c , ( 2nd ` ( Q ` c ) ) >. )
26	25	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( a e. om /\ c e. om ) -> ( Q ` c ) = <. c , ( 2nd ` ( Q ` c ) ) >. )
27	26	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. <-> <. c , ( 2nd ` ( Q ` c ) ) >. = <. a , b >. ) )
28		vex 3203	. . . . . . . . . . . . . . 15 |- c e. _V
29		fvex 6201	. . . . . . . . . . . . . . 15 |- ( 2nd ` ( Q ` c ) ) e. _V
30	28, 29	opth1 4944	. . . . . . . . . . . . . 14 |- ( <. c , ( 2nd ` ( Q ` c ) ) >. = <. a , b >. -> c = a )
31	27, 30	syl6bi 243	. . . . . . . . . . . . 13 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. -> c = a ) )
32		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( c = a -> ( Q ` c ) = ( Q ` a ) )
33	32	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( c = a -> ( ( Q ` c ) = <. a , b >. <-> ( Q ` a ) = <. a , b >. ) )
34	33	biimpd 219	. . . . . . . . . . . . 13 |- ( c = a -> ( ( Q ` c ) = <. a , b >. -> ( Q ` a ) = <. a , b >. ) )
35	31, 34	syli 39	. . . . . . . . . . . 12 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. -> ( Q ` a ) = <. a , b >. ) )
36		fveq2 6191	. . . . . . . . . . . . 13 |- ( ( Q ` a ) = <. a , b >. -> ( 2nd ` ( Q ` a ) ) = ( 2nd ` <. a , b >. ) )
37		vex 3203	. . . . . . . . . . . . . 14 |- a e. _V
38		vex 3203	. . . . . . . . . . . . . 14 |- b e. _V
39	37, 38	op2nd 7177	. . . . . . . . . . . . 13 |- ( 2nd ` <. a , b >. ) = b
40	36, 39	syl6req 2673	. . . . . . . . . . . 12 |- ( ( Q ` a ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) )
41	35, 40	syl6 35	. . . . . . . . . . 11 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) ) )
42	41	rexlimdva 3031	. . . . . . . . . 10 |- ( a e. om -> ( E. c e. om ( Q ` c ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) ) )
43	2	seqomlem1 7545	. . . . . . . . . . . 12 |- ( a e. om -> ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
44	32	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( c = a -> ( ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. <-> ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) )
45	44	rspcev 3309	. . . . . . . . . . . 12 |- ( ( a e. om /\ ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) -> E. c e. om ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
46	43, 45	mpdan 702	. . . . . . . . . . 11 |- ( a e. om -> E. c e. om ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
47		opeq2 4403	. . . . . . . . . . . . 13 |- ( b = ( 2nd ` ( Q ` a ) ) -> <. a , b >. = <. a , ( 2nd ` ( Q ` a ) ) >. )
48	47	eqeq2d 2632	. . . . . . . . . . . 12 |- ( b = ( 2nd ` ( Q ` a ) ) -> ( ( Q ` c ) = <. a , b >. <-> ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) )
49	48	rexbidv 3052	. . . . . . . . . . 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 237	. . . . . . . . . 10 |- ( a e. om -> ( b = ( 2nd ` ( Q ` a ) ) -> E. c e. om ( Q ` c ) = <. a , b >. ) )
51	42, 50	impbid 202	. . . . . . . . 9 |- ( a e. om -> ( E. c e. om ( Q ` c ) = <. a , b >. <-> b = ( 2nd ` ( Q ` a ) ) ) )
52	24, 51	syl5bb 272	. . . . . . . 8 |- ( a e. om -> ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) )
53	52	alrimiv 1855	. . . . . . 7 |- ( a e. om -> A. b ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) )
54		fvex 6201	. . . . . . . 8 |- ( 2nd ` ( Q ` a ) ) e. _V
55		eqeq2 2633	. . . . . . . . . 10 |- ( c = ( 2nd ` ( Q ` a ) ) -> ( b = c <-> b = ( 2nd ` ( Q ` a ) ) ) )
56	55	bibi2d 332	. . . . . . . . 9 |- ( c = ( 2nd ` ( Q ` a ) ) -> ( ( a ran ( Q |` om ) b <-> b = c ) <-> ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) ) )
57	56	albidv 1849	. . . . . . . 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 3300	. . . . . . 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 17	. . . . . 6 |- ( a e. om -> E. c A. b ( a ran ( Q |` om ) b <-> b = c ) )
60		df-eu 2474	. . . . . 6 |- ( E! b a ran ( Q |` om ) b <-> E. c A. b ( a ran ( Q |` om ) b <-> b = c ) )
61	59, 60	sylibr 224	. . . . 5 |- ( a e. om -> E! b a ran ( Q |` om ) b )
62	61	rgen 2922	. . . 4 |- A. a e. om E! b a ran ( Q |` om ) b
63		dff3 6372	. . . 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 955	. . 3 |- ran ( Q |` om ) : om --> _V
65		df-ima 5127	. . . 4 |- ( Q " om ) = ran ( Q |` om )
66	65	feq1i 6036	. . 3 |- ( ( Q " om ) : om --> _V <-> ran ( Q |` om ) : om --> _V )
67	64, 66	mpbir 221	. 2 |- ( Q " om ) : om --> _V
68		dffn2 6047	. 2 |- ( ( Q " om ) Fn om <-> ( Q " om ) : om --> _V )
69	67, 68	mpbir 221	1 |- ( Q " om ) Fn om

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   <.cop 4183   class class class wbr 4653   _I cid 5023   X. cxp 5112   ran crn 5115   |` cres 5116   "cima 5117   suc csuc 5725   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   2ndc2nd 7167   reccrdg 7505

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

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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  seqomlem3  7547  seqomlem4  7548  fnseqom  7550