Theorem seqomlem4 7548

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
seqomlem4	|- ( A e. om -> ( ( Q " om ) ` suc A ) = ( A F ( ( Q " om ) ` A ) ) )

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

Allowed substitution hints:   I( v, i)

Proof of Theorem seqomlem4

Step	Hyp	Ref	Expression
1		peano2 7086	. . . . . . 7 |- ( A e. om -> suc A e. om )
2		fvres 6207	. . . . . . 7 |- ( suc A e. om -> ( ( Q |` om ) ` suc A ) = ( Q ` suc A ) )
3	1, 2	syl 17	. . . . . 6 |- ( A e. om -> ( ( Q |` om ) ` suc A ) = ( Q ` suc A ) )
4		frsuc 7532	. . . . . . . 8 |- ( A e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` suc A ) = ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` A ) ) )
5		fvres 6207	. . . . . . . . . 10 |- ( suc A e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` suc A ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` suc A ) )
6	1, 5	syl 17	. . . . . . . . 9 |- ( A e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` suc A ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` suc A ) )
7		seqomlem.a	. . . . . . . . . 10 |- Q = rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. )
8	7	fveq1i 6192	. . . . . . . . 9 |- ( Q ` suc A ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` suc A )
9	6, 8	syl6eqr 2674	. . . . . . . 8 |- ( A e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` suc A ) = ( Q ` suc A ) )
10		fvres 6207	. . . . . . . . . 10 |- ( A e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` A ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` A ) )
11	7	fveq1i 6192	. . . . . . . . . 10 |- ( Q ` A ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` A )
12	10, 11	syl6eqr 2674	. . . . . . . . 9 |- ( A e. om -> ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` A ) = ( Q ` A ) )
13	12	fveq2d 6195	. . . . . . . 8 |- ( A e. om -> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) ` A ) ) = ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` A ) ) )
14	4, 9, 13	3eqtr3d 2664	. . . . . . 7 |- ( A e. om -> ( Q ` suc A ) = ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` A ) ) )
15	7	seqomlem1 7545	. . . . . . . 8 |- ( A e. om -> ( Q ` A ) = <. A , ( 2nd ` ( Q ` A ) ) >. )
16	15	fveq2d 6195	. . . . . . 7 |- ( A e. om -> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` A ) ) = ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` <. A , ( 2nd ` ( Q ` A ) ) >. ) )
17		df-ov 6653	. . . . . . . 8 |- ( A ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` A ) ) ) = ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` <. A , ( 2nd ` ( Q ` A ) ) >. )
18		fvex 6201	. . . . . . . . . 10 |- ( 2nd ` ( Q ` A ) ) e. _V
19		suceq 5790	. . . . . . . . . . . 12 |- ( i = A -> suc i = suc A )
20		oveq1 6657	. . . . . . . . . . . 12 |- ( i = A -> ( i F v ) = ( A F v ) )
21	19, 20	opeq12d 4410	. . . . . . . . . . 11 |- ( i = A -> <. suc i , ( i F v ) >. = <. suc A , ( A F v ) >. )
22		oveq2 6658	. . . . . . . . . . . 12 |- ( v = ( 2nd ` ( Q ` A ) ) -> ( A F v ) = ( A F ( 2nd ` ( Q ` A ) ) ) )
23	22	opeq2d 4409	. . . . . . . . . . 11 |- ( v = ( 2nd ` ( Q ` A ) ) -> <. suc A , ( A F v ) >. = <. suc A , ( A F ( 2nd ` ( Q ` A ) ) ) >. )
24		eqid 2622	. . . . . . . . . . 11 |- ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) = ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. )
25		opex 4932	. . . . . . . . . . 11 |- <. suc A , ( A F ( 2nd ` ( Q ` A ) ) ) >. e. _V
26	21, 23, 24, 25	ovmpt2 6796	. . . . . . . . . 10 |- ( ( A e. om /\ ( 2nd ` ( Q ` A ) ) e. _V ) -> ( A ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` A ) ) ) = <. suc A , ( A F ( 2nd ` ( Q ` A ) ) ) >. )
27	18, 26	mpan2 707	. . . . . . . . 9 |- ( A e. om -> ( A ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` A ) ) ) = <. suc A , ( A F ( 2nd ` ( Q ` A ) ) ) >. )
28		fvres 6207	. . . . . . . . . . . . . . . . 17 |- ( A e. om -> ( ( Q |` om ) ` A ) = ( Q ` A ) )
29	28, 15	eqtrd 2656	. . . . . . . . . . . . . . . 16 |- ( A e. om -> ( ( Q |` om ) ` A ) = <. A , ( 2nd ` ( Q ` A ) ) >. )
30		frfnom 7530	. . . . . . . . . . . . . . . . . 18 |- ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) Fn om
31	7	reseq1i 5392	. . . . . . . . . . . . . . . . . . 19 |- ( Q |` om ) = ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om )
32	31	fneq1i 5985	. . . . . . . . . . . . . . . . . 18 |- ( ( Q |` om ) Fn om <-> ( rec ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` om ) Fn om )
33	30, 32	mpbir 221	. . . . . . . . . . . . . . . . 17 |- ( Q |` om ) Fn om
34		fnfvelrn 6356	. . . . . . . . . . . . . . . . 17 |- ( ( ( Q |` om ) Fn om /\ A e. om ) -> ( ( Q |` om ) ` A ) e. ran ( Q |` om ) )
35	33, 34	mpan 706	. . . . . . . . . . . . . . . 16 |- ( A e. om -> ( ( Q |` om ) ` A ) e. ran ( Q |` om ) )
36	29, 35	eqeltrrd 2702	. . . . . . . . . . . . . . 15 |- ( A e. om -> <. A , ( 2nd ` ( Q ` A ) ) >. e. ran ( Q |` om ) )
37		df-ima 5127	. . . . . . . . . . . . . . 15 |- ( Q " om ) = ran ( Q |` om )
38	36, 37	syl6eleqr 2712	. . . . . . . . . . . . . 14 |- ( A e. om -> <. A , ( 2nd ` ( Q ` A ) ) >. e. ( Q " om ) )
39		df-br 4654	. . . . . . . . . . . . . 14 |- ( A ( Q " om ) ( 2nd ` ( Q ` A ) ) <-> <. A , ( 2nd ` ( Q ` A ) ) >. e. ( Q " om ) )
40	38, 39	sylibr 224	. . . . . . . . . . . . 13 |- ( A e. om -> A ( Q " om ) ( 2nd ` ( Q ` A ) ) )
41	7	seqomlem2 7546	. . . . . . . . . . . . . 14 |- ( Q " om ) Fn om
42		fnbrfvb 6236	. . . . . . . . . . . . . 14 |- ( ( ( Q " om ) Fn om /\ A e. om ) -> ( ( ( Q " om ) ` A ) = ( 2nd ` ( Q ` A ) ) <-> A ( Q " om ) ( 2nd ` ( Q ` A ) ) ) )
43	41, 42	mpan 706	. . . . . . . . . . . . 13 |- ( A e. om -> ( ( ( Q " om ) ` A ) = ( 2nd ` ( Q ` A ) ) <-> A ( Q " om ) ( 2nd ` ( Q ` A ) ) ) )
44	40, 43	mpbird 247	. . . . . . . . . . . 12 |- ( A e. om -> ( ( Q " om ) ` A ) = ( 2nd ` ( Q ` A ) ) )
45	44	eqcomd 2628	. . . . . . . . . . 11 |- ( A e. om -> ( 2nd ` ( Q ` A ) ) = ( ( Q " om ) ` A ) )
46	45	oveq2d 6666	. . . . . . . . . 10 |- ( A e. om -> ( A F ( 2nd ` ( Q ` A ) ) ) = ( A F ( ( Q " om ) ` A ) ) )
47	46	opeq2d 4409	. . . . . . . . 9 |- ( A e. om -> <. suc A , ( A F ( 2nd ` ( Q ` A ) ) ) >. = <. suc A , ( A F ( ( Q " om ) ` A ) ) >. )
48	27, 47	eqtrd 2656	. . . . . . . 8 |- ( A e. om -> ( A ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` A ) ) ) = <. suc A , ( A F ( ( Q " om ) ` A ) ) >. )
49	17, 48	syl5eqr 2670	. . . . . . 7 |- ( A e. om -> ( ( i e. om , v e. _V |-> <. suc i , ( i F v ) >. ) ` <. A , ( 2nd ` ( Q ` A ) ) >. ) = <. suc A , ( A F ( ( Q " om ) ` A ) ) >. )
50	14, 16, 49	3eqtrd 2660	. . . . . 6 |- ( A e. om -> ( Q ` suc A ) = <. suc A , ( A F ( ( Q " om ) ` A ) ) >. )
51	3, 50	eqtrd 2656	. . . . 5 |- ( A e. om -> ( ( Q |` om ) ` suc A ) = <. suc A , ( A F ( ( Q " om ) ` A ) ) >. )
52		fnfvelrn 6356	. . . . . 6 |- ( ( ( Q |` om ) Fn om /\ suc A e. om ) -> ( ( Q |` om ) ` suc A ) e. ran ( Q |` om ) )
53	33, 1, 52	sylancr 695	. . . . 5 |- ( A e. om -> ( ( Q |` om ) ` suc A ) e. ran ( Q |` om ) )
54	51, 53	eqeltrrd 2702	. . . 4 |- ( A e. om -> <. suc A , ( A F ( ( Q " om ) ` A ) ) >. e. ran ( Q |` om ) )
55	54, 37	syl6eleqr 2712	. . 3 |- ( A e. om -> <. suc A , ( A F ( ( Q " om ) ` A ) ) >. e. ( Q " om ) )
56		df-br 4654	. . 3 |- ( suc A ( Q " om ) ( A F ( ( Q " om ) ` A ) ) <-> <. suc A , ( A F ( ( Q " om ) ` A ) ) >. e. ( Q " om ) )
57	55, 56	sylibr 224	. 2 |- ( A e. om -> suc A ( Q " om ) ( A F ( ( Q " om ) ` A ) ) )
58		fnbrfvb 6236	. . 3 |- ( ( ( Q " om ) Fn om /\ suc A e. om ) -> ( ( ( Q " om ) ` suc A ) = ( A F ( ( Q " om ) ` A ) ) <-> suc A ( Q " om ) ( A F ( ( Q " om ) ` A ) ) ) )
59	41, 1, 58	sylancr 695	. 2 |- ( A e. om -> ( ( ( Q " om ) ` suc A ) = ( A F ( ( Q " om ) ` A ) ) <-> suc A ( Q " om ) ( A F ( ( Q " om ) ` A ) ) ) )
60	57, 59	mpbird 247	1 |- ( A e. om -> ( ( Q " om ) ` suc A ) = ( A F ( ( Q " om ) ` A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   <.cop 4183   class class class wbr 4653   _I cid 5023   ran crn 5115   |` cres 5116   "cima 5117   suc csuc 5725   Fn wfn 5883   ` 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:  seqomsuc  7552