Theorem seqomlem2 7123

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 7107	. . . . . . 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 5063	. . . . . . . 8 |- ( Q |` om ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om )
4	3	fneq1i 5631	. . . . . . 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 212	. . . . . 6 |- ( Q |` om ) Fn om
6		fvres 5839	. . . . . . . . 9 |- ( b e. om -> ( ( Q |` om ) ` b ) = ( Q ` b ) )
7	2	seqomlem1 7122	. . . . . . . . 9 |- ( b e. om -> ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. )
8	6, 7	eqtrd 2462	. . . . . . . 8 |- ( b e. om -> ( ( Q |` om ) ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. )
9		fvex 5835	. . . . . . . . 9 |- ( 2nd ` ( Q ` b ) ) e. _V
10		opelxpi 4828	. . . . . . . . 9 |- ( ( b e. om /\ ( 2nd ` ( Q ` b ) ) e. _V ) -> <. b , ( 2nd ` ( Q ` b ) ) >. e. ( om X. _V ) )
11	9, 10	mpan2 675	. . . . . . . 8 |- ( b e. om -> <. b , ( 2nd ` ( Q ` b ) ) >. e. ( om X. _V ) )
12	8, 11	eqeltrd 2506	. . . . . . 7 |- ( b e. om -> ( ( Q |` om ) ` b ) e. ( om X. _V ) )
13	12	rgen 2724	. . . . . 6 |- A. b e. om ( ( Q |` om ) ` b ) e. ( om X. _V )
14		ffnfv 6008	. . . . . 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 928	. . . . 5 |- ( Q |` om ) : om --> ( om X. _V )
16		frn 5695	. . . . 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 4367	. . . . . . . . . 10 |- ( a ran ( Q |` om ) b <-> <. a , b >. e. ran ( Q |` om ) )
19		fvelrnb 5872	. . . . . . . . . . 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 5839	. . . . . . . . . . . 12 |- ( c e. om -> ( ( Q |` om ) ` c ) = ( Q ` c ) )
22	21	eqeq1d 2430	. . . . . . . . . . 11 |- ( c e. om -> ( ( ( Q |` om ) ` c ) = <. a , b >. <-> ( Q ` c ) = <. a , b >. ) )
23	22	rexbiia 2865	. . . . . . . . . 10 |- ( E. c e. om ( ( Q |` om ) ` c ) = <. a , b >. <-> E. c e. om ( Q ` c ) = <. a , b >. )
24	18, 20, 23	3bitri 274	. . . . . . . . 9 |- ( a ran ( Q |` om ) b <-> E. c e. om ( Q ` c ) = <. a , b >. )
25	2	seqomlem1 7122	. . . . . . . . . . . . . . . 16 |- ( c e. om -> ( Q ` c ) = <. c , ( 2nd ` ( Q ` c ) ) >. )
26	25	adantl 467	. . . . . . . . . . . . . . 15 |- ( ( a e. om /\ c e. om ) -> ( Q ` c ) = <. c , ( 2nd ` ( Q ` c ) ) >. )
27	26	eqeq1d 2430	. . . . . . . . . . . . . 14 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. <-> <. c , ( 2nd ` ( Q ` c ) ) >. = <. a , b >. ) )
28		vex 3025	. . . . . . . . . . . . . . 15 |- c e. _V
29		fvex 5835	. . . . . . . . . . . . . . 15 |- ( 2nd ` ( Q ` c ) ) e. _V
30	28, 29	opth1 4637	. . . . . . . . . . . . . 14 |- ( <. c , ( 2nd ` ( Q ` c ) ) >. = <. a , b >. -> c = a )
31	27, 30	syl6bi 231	. . . . . . . . . . . . 13 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. -> c = a ) )
32		fveq2 5825	. . . . . . . . . . . . . . 15 |- ( c = a -> ( Q ` c ) = ( Q ` a ) )
33	32	eqeq1d 2430	. . . . . . . . . . . . . 14 |- ( c = a -> ( ( Q ` c ) = <. a , b >. <-> ( Q ` a ) = <. a , b >. ) )
34	33	biimpd 210	. . . . . . . . . . . . 13 |- ( c = a -> ( ( Q ` c ) = <. a , b >. -> ( Q ` a ) = <. a , b >. ) )
35	31, 34	syli 38	. . . . . . . . . . . 12 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. -> ( Q ` a ) = <. a , b >. ) )
36		fveq2 5825	. . . . . . . . . . . . 13 |- ( ( Q ` a ) = <. a , b >. -> ( 2nd ` ( Q ` a ) ) = ( 2nd ` <. a , b >. ) )
37		vex 3025	. . . . . . . . . . . . . 14 |- a e. _V
38		vex 3025	. . . . . . . . . . . . . 14 |- b e. _V
39	37, 38	op2nd 6760	. . . . . . . . . . . . 13 |- ( 2nd ` <. a , b >. ) = b
40	36, 39	syl6req 2479	. . . . . . . . . . . 12 |- ( ( Q ` a ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) )
41	35, 40	syl6 34	. . . . . . . . . . 11 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) ) )
42	41	rexlimdva 2856	. . . . . . . . . 10 |- ( a e. om -> ( E. c e. om ( Q ` c ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) ) )
43	2	seqomlem1 7122	. . . . . . . . . . . 12 |- ( a e. om -> ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
44	32	eqeq1d 2430	. . . . . . . . . . . . 13 |- ( c = a -> ( ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. <-> ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) )
45	44	rspcev 3125	. . . . . . . . . . . 12 |- ( ( a e. om /\ ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) -> E. c e. om ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
46	43, 45	mpdan 672	. . . . . . . . . . 11 |- ( a e. om -> E. c e. om ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
47		opeq2 4131	. . . . . . . . . . . . 13 |- ( b = ( 2nd ` ( Q ` a ) ) -> <. a , b >. = <. a , ( 2nd ` ( Q ` a ) ) >. )
48	47	eqeq2d 2438	. . . . . . . . . . . 12 |- ( b = ( 2nd ` ( Q ` a ) ) -> ( ( Q ` c ) = <. a , b >. <-> ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) )
49	48	rexbidv 2878	. . . . . . . . . . 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 225	. . . . . . . . . 10 |- ( a e. om -> ( b = ( 2nd ` ( Q ` a ) ) -> E. c e. om ( Q ` c ) = <. a , b >. ) )
51	42, 50	impbid 193	. . . . . . . . 9 |- ( a e. om -> ( E. c e. om ( Q ` c ) = <. a , b >. <-> b = ( 2nd ` ( Q ` a ) ) ) )
52	24, 51	syl5bb 260	. . . . . . . 8 |- ( a e. om -> ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) )
53	52	alrimiv 1767	. . . . . . 7 |- ( a e. om -> A. b ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) )
54		fvex 5835	. . . . . . . 8 |- ( 2nd ` ( Q ` a ) ) e. _V
55		eqeq2 2439	. . . . . . . . . 10 |- ( c = ( 2nd ` ( Q ` a ) ) -> ( b = c <-> b = ( 2nd ` ( Q ` a ) ) ) )
56	55	bibi2d 319	. . . . . . . . 9 |- ( c = ( 2nd ` ( Q ` a ) ) -> ( ( a ran ( Q |` om ) b <-> b = c ) <-> ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) ) )
57	56	albidv 1761	. . . . . . . 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 3116	. . . . . . 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 2280	. . . . . 6 |- ( E! b a ran ( Q |` om ) b <-> E. c A. b ( a ran ( Q |` om ) b <-> b = c ) )
61	59, 60	sylibr 215	. . . . 5 |- ( a e. om -> E! b a ran ( Q |` om ) b )
62	61	rgen 2724	. . . 4 |- A. a e. om E! b a ran ( Q |` om ) b
63		dff3 5994	. . . 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 928	. . 3 |- ran ( Q |` om ) : om --> _V
65		df-ima 4809	. . . 4 |- ( Q " om ) = ran ( Q |` om )
66	65	feq1i 5681	. . 3 |- ( ( Q " om ) : om --> _V <-> ran ( Q |` om ) : om --> _V )
67	64, 66	mpbir 212	. 2 |- ( Q " om ) : om --> _V
68		dffn2 5690	. 2 |- ( ( Q " om ) Fn om <-> ( Q " om ) : om --> _V )
69	67, 68	mpbir 212	1 |- ( Q " om ) Fn om

Syntax hints:   <-> wb 187   /\ wa 370   A.wal 1435   = wceq 1437   E.wex 1657   e. wcel 1872   E!weu 2276   A.wral 2714   E.wrex 2715   _Vcvv 3022   C_ wss 3379   (/)c0 3704   <.cop 3947   class class class wbr 4366   _I cid 4706   X. cxp 4794   ran crn 4797   |` cres 4798   "cima 4799   suc csuc 5387   Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249   |-> cmpt2 6251   omcom 6650   2ndc2nd 6750   reccrdg 7082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083

This theorem is referenced by:  seqomlem3  7124  seqomlem4  7125  fnseqom  7127