Theorem seqomlem2 7186

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 7170	. . . . . . 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 5107	. . . . . . . 8 |- ( Q |` om ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om )
4	3	fneq1i 5680	. . . . . . 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 214	. . . . . 6 |- ( Q |` om ) Fn om
6		fvres 5893	. . . . . . . . 9 |- ( b e. om -> ( ( Q |` om ) ` b ) = ( Q ` b ) )
7	2	seqomlem1 7185	. . . . . . . . 9 |- ( b e. om -> ( Q ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. )
8	6, 7	eqtrd 2505	. . . . . . . 8 |- ( b e. om -> ( ( Q |` om ) ` b ) = <. b , ( 2nd ` ( Q ` b ) ) >. )
9		fvex 5889	. . . . . . . . 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 685	. . . . . . . 8 |- ( b e. om -> <. b , ( 2nd ` ( Q ` b ) ) >. e. ( om X. _V ) )
12	8, 11	eqeltrd 2549	. . . . . . 7 |- ( b e. om -> ( ( Q |` om ) ` b ) e. ( om X. _V ) )
13	12	rgen 2766	. . . . . 6 |- A. b e. om ( ( Q |` om ) ` b ) e. ( om X. _V )
14		ffnfv 6064	. . . . . 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 934	. . . . 5 |- ( Q |` om ) : om --> ( om X. _V )
16		frn 5747	. . . . 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 4396	. . . . . . . . . 10 |- ( a ran ( Q |` om ) b <-> <. a , b >. e. ran ( Q |` om ) )
19		fvelrnb 5926	. . . . . . . . . . 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 5893	. . . . . . . . . . . 12 |- ( c e. om -> ( ( Q |` om ) ` c ) = ( Q ` c ) )
22	21	eqeq1d 2473	. . . . . . . . . . 11 |- ( c e. om -> ( ( ( Q |` om ) ` c ) = <. a , b >. <-> ( Q ` c ) = <. a , b >. ) )
23	22	rexbiia 2880	. . . . . . . . . 10 |- ( E. c e. om ( ( Q |` om ) ` c ) = <. a , b >. <-> E. c e. om ( Q ` c ) = <. a , b >. )
24	18, 20, 23	3bitri 279	. . . . . . . . 9 |- ( a ran ( Q |` om ) b <-> E. c e. om ( Q ` c ) = <. a , b >. )
25	2	seqomlem1 7185	. . . . . . . . . . . . . . . 16 |- ( c e. om -> ( Q ` c ) = <. c , ( 2nd ` ( Q ` c ) ) >. )
26	25	adantl 473	. . . . . . . . . . . . . . 15 |- ( ( a e. om /\ c e. om ) -> ( Q ` c ) = <. c , ( 2nd ` ( Q ` c ) ) >. )
27	26	eqeq1d 2473	. . . . . . . . . . . . . 14 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. <-> <. c , ( 2nd ` ( Q ` c ) ) >. = <. a , b >. ) )
28		vex 3034	. . . . . . . . . . . . . . 15 |- c e. _V
29		fvex 5889	. . . . . . . . . . . . . . 15 |- ( 2nd ` ( Q ` c ) ) e. _V
30	28, 29	opth1 4675	. . . . . . . . . . . . . 14 |- ( <. c , ( 2nd ` ( Q ` c ) ) >. = <. a , b >. -> c = a )
31	27, 30	syl6bi 236	. . . . . . . . . . . . 13 |- ( ( a e. om /\ c e. om ) -> ( ( Q ` c ) = <. a , b >. -> c = a ) )
32		fveq2 5879	. . . . . . . . . . . . . . 15 |- ( c = a -> ( Q ` c ) = ( Q ` a ) )
33	32	eqeq1d 2473	. . . . . . . . . . . . . 14 |- ( c = a -> ( ( Q ` c ) = <. a , b >. <-> ( Q ` a ) = <. a , b >. ) )
34	33	biimpd 212	. . . . . . . . . . . . 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 5879	. . . . . . . . . . . . 13 |- ( ( Q ` a ) = <. a , b >. -> ( 2nd ` ( Q ` a ) ) = ( 2nd ` <. a , b >. ) )
37		vex 3034	. . . . . . . . . . . . . 14 |- a e. _V
38		vex 3034	. . . . . . . . . . . . . 14 |- b e. _V
39	37, 38	op2nd 6821	. . . . . . . . . . . . 13 |- ( 2nd ` <. a , b >. ) = b
40	36, 39	syl6req 2522	. . . . . . . . . . . 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 2871	. . . . . . . . . 10 |- ( a e. om -> ( E. c e. om ( Q ` c ) = <. a , b >. -> b = ( 2nd ` ( Q ` a ) ) ) )
43	2	seqomlem1 7185	. . . . . . . . . . . 12 |- ( a e. om -> ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
44	32	eqeq1d 2473	. . . . . . . . . . . . 13 |- ( c = a -> ( ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. <-> ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) )
45	44	rspcev 3136	. . . . . . . . . . . 12 |- ( ( a e. om /\ ( Q ` a ) = <. a , ( 2nd ` ( Q ` a ) ) >. ) -> E. c e. om ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
46	43, 45	mpdan 681	. . . . . . . . . . 11 |- ( a e. om -> E. c e. om ( Q ` c ) = <. a , ( 2nd ` ( Q ` a ) ) >. )
47		opeq2 4159	. . . . . . . . . . . . 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 2892	. . . . . . . . . . 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 230	. . . . . . . . . 10 |- ( a e. om -> ( b = ( 2nd ` ( Q ` a ) ) -> E. c e. om ( Q ` c ) = <. a , b >. ) )
51	42, 50	impbid 195	. . . . . . . . 9 |- ( a e. om -> ( E. c e. om ( Q ` c ) = <. a , b >. <-> b = ( 2nd ` ( Q ` a ) ) ) )
52	24, 51	syl5bb 265	. . . . . . . 8 |- ( a e. om -> ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) )
53	52	alrimiv 1781	. . . . . . 7 |- ( a e. om -> A. b ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) )
54		fvex 5889	. . . . . . . 8 |- ( 2nd ` ( Q ` a ) ) e. _V
55		eqeq2 2482	. . . . . . . . . 10 |- ( c = ( 2nd ` ( Q ` a ) ) -> ( b = c <-> b = ( 2nd ` ( Q ` a ) ) ) )
56	55	bibi2d 325	. . . . . . . . 9 |- ( c = ( 2nd ` ( Q ` a ) ) -> ( ( a ran ( Q |` om ) b <-> b = c ) <-> ( a ran ( Q |` om ) b <-> b = ( 2nd ` ( Q ` a ) ) ) ) )
57	56	albidv 1775	. . . . . . . 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 3127	. . . . . . 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 2323	. . . . . 6 |- ( E! b a ran ( Q |` om ) b <-> E. c A. b ( a ran ( Q |` om ) b <-> b = c ) )
61	59, 60	sylibr 217	. . . . 5 |- ( a e. om -> E! b a ran ( Q |` om ) b )
62	61	rgen 2766	. . . 4 |- A. a e. om E! b a ran ( Q |` om ) b
63		dff3 6050	. . . 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 934	. . 3 |- ran ( Q |` om ) : om --> _V
65		df-ima 4852	. . . 4 |- ( Q " om ) = ran ( Q |` om )
66	65	feq1i 5730	. . 3 |- ( ( Q " om ) : om --> _V <-> ran ( Q |` om ) : om --> _V )
67	64, 66	mpbir 214	. 2 |- ( Q " om ) : om --> _V
68		dffn2 5741	. 2 |- ( ( Q " om ) Fn om <-> ( Q " om ) : om --> _V )
69	67, 68	mpbir 214	1 |- ( Q " om ) Fn om

Syntax hints:   <-> wb 189   /\ wa 376   A.wal 1450   = wceq 1452   E.wex 1671   e. wcel 1904   E!weu 2319   A.wral 2756   E.wrex 2757   _Vcvv 3031   C_ wss 3390   (/)c0 3722   <.cop 3965   class class class wbr 4395   _I cid 4749   X. cxp 4837   ran crn 4840   |` cres 4841   "cima 4842   suc csuc 5432   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   omcom 6711   2ndc2nd 6811   reccrdg 7145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146

This theorem is referenced by:  seqomlem3  7187  seqomlem4  7188  fnseqom  7190