Theorem tfrlem11 7047

Description: Lemma for transfinite recursion. Compute the value of C. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypotheses

Ref	Expression
tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
tfrlem.3	|- C = (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )

Assertion

Ref	Expression
tfrlem11	|- ( dom recs ( F ) e. On -> ( B e. suc dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )

Distinct variable groups:   x, f, y, B   C, f, x, y   f, F, x, y

Allowed substitution hints:   A( x, y, f)

Proof of Theorem tfrlem11

Step	Hyp	Ref	Expression
1		elsuci 4937	. 2 |- ( B e. suc dom recs ( F ) -> ( B e. dom recs ( F ) \/ B = dom recs ( F ) ) )
2		tfrlem.1	. . . . . . . . 9 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
3		tfrlem.3	. . . . . . . . 9 |- C = (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
4	2, 3	tfrlem10 7046	. . . . . . . 8 |- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) )
5		fnfun 5669	. . . . . . . 8 |- ( C Fn suc dom recs ( F ) -> Fun C )
6	4, 5	syl 16	. . . . . . 7 |- ( dom recs ( F ) e. On -> Fun C )
7		ssun1 3660	. . . . . . . . 9 |- recs ( F ) C_ (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
8	7, 3	sseqtr4i 3530	. . . . . . . 8 |- recs ( F ) C_ C
9	2	tfrlem9 7044	. . . . . . . . 9 |- ( B e. dom recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
10		funssfv 5872	. . . . . . . . . . . 12 |- ( ( Fun C /\ recs ( F ) C_ C /\ B e. dom recs ( F ) ) -> ( C ` B ) = (recs ( F ) ` B ) )
11	10	3expa 1191	. . . . . . . . . . 11 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B e. dom recs ( F ) ) -> ( C ` B ) = (recs ( F ) ` B ) )
12	11	adantrl 715	. . . . . . . . . 10 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( C ` B ) = (recs ( F ) ` B ) )
13		onelss 4913	. . . . . . . . . . . 12 |- ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> B C_ dom recs ( F ) ) )
14	13	imp 429	. . . . . . . . . . 11 |- ( ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) -> B C_ dom recs ( F ) )
15		fun2ssres 5620	. . . . . . . . . . . . 13 |- ( ( Fun C /\ recs ( F ) C_ C /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
16	15	3expa 1191	. . . . . . . . . . . 12 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
17	16	fveq2d 5861	. . . . . . . . . . 11 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B C_ dom recs ( F ) ) -> ( F ` ( C |` B ) ) = ( F ` (recs ( F ) |` B ) ) )
18	14, 17	sylan2 474	. . . . . . . . . 10 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( F ` ( C |` B ) ) = ( F ` (recs ( F ) |` B ) ) )
19	12, 18	eqeq12d 2482	. . . . . . . . 9 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( ( C ` B ) = ( F ` ( C |` B ) ) <-> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) )
20	9, 19	syl5ibr 221	. . . . . . . 8 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
21	8, 20	mpanl2 681	. . . . . . 7 |- ( ( Fun C /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
22	6, 21	sylan 471	. . . . . 6 |- ( ( dom recs ( F ) e. On /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
23	22	exp32 605	. . . . 5 |- ( dom recs ( F ) e. On -> ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) ) )
24	23	pm2.43i 47	. . . 4 |- ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) )
25	24	pm2.43d 48	. . 3 |- ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
26		opex 4704	. . . . . . . . 9 |- <. B , ( F ` ( C |` B ) ) >. e. _V
27	26	snid 4048	. . . . . . . 8 |- <. B , ( F ` ( C |` B ) ) >. e. { <. B , ( F ` ( C |` B ) ) >. }
28		opeq1 4206	. . . . . . . . . . 11 |- ( B = dom recs ( F ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` ( C |` B ) ) >. )
29	28	adantl 466	. . . . . . . . . 10 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` ( C |` B ) ) >. )
30		eqimss 3549	. . . . . . . . . . . . . 14 |- ( B = dom recs ( F ) -> B C_ dom recs ( F ) )
31	8, 15	mp3an2 1307	. . . . . . . . . . . . . 14 |- ( ( Fun C /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
32	6, 30, 31	syl2an 477	. . . . . . . . . . . . 13 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
33		reseq2 5259	. . . . . . . . . . . . . . 15 |- ( B = dom recs ( F ) -> (recs ( F ) |` B ) = (recs ( F ) |` dom recs ( F ) ) )
34	2	tfrlem6 7041	. . . . . . . . . . . . . . . 16 |- Rel recs ( F )
35		resdm 5306	. . . . . . . . . . . . . . . 16 |- ( Rel recs ( F ) -> (recs ( F ) |` dom recs ( F ) ) = recs ( F ) )
36	34, 35	ax-mp 5	. . . . . . . . . . . . . . 15 |- (recs ( F ) |` dom recs ( F ) ) = recs ( F )
37	33, 36	syl6eq 2517	. . . . . . . . . . . . . 14 |- ( B = dom recs ( F ) -> (recs ( F ) |` B ) = recs ( F ) )
38	37	adantl 466	. . . . . . . . . . . . 13 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> (recs ( F ) |` B ) = recs ( F ) )
39	32, 38	eqtrd 2501	. . . . . . . . . . . 12 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C |` B ) = recs ( F ) )
40	39	fveq2d 5861	. . . . . . . . . . 11 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( F ` ( C |` B ) ) = ( F ` recs ( F ) ) )
41	40	opeq2d 4213	. . . . . . . . . 10 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. dom recs ( F ) , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` recs ( F ) ) >. )
42	29, 41	eqtrd 2501	. . . . . . . . 9 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` recs ( F ) ) >. )
43	42	sneqd 4032	. . . . . . . 8 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> { <. B , ( F ` ( C |` B ) ) >. } = { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
44	27, 43	syl5eleq 2554	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
45		elun2 3665	. . . . . . 7 |- ( <. B , ( F ` ( C |` B ) ) >. e. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } -> <. B , ( F ` ( C |` B ) ) >. e. (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) )
46	44, 45	syl 16	. . . . . 6 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) )
47	46, 3	syl6eleqr 2559	. . . . 5 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. C )
48	4	adantr 465	. . . . . 6 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> C Fn suc dom recs ( F ) )
49		simpr 461	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> B = dom recs ( F ) )
50		sucidg 4949	. . . . . . . 8 |- ( dom recs ( F ) e. On -> dom recs ( F ) e. suc dom recs ( F ) )
51	50	adantr 465	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> dom recs ( F ) e. suc dom recs ( F ) )
52	49, 51	eqeltrd 2548	. . . . . 6 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> B e. suc dom recs ( F ) )
53		fnopfvb 5900	. . . . . 6 |- ( ( C Fn suc dom recs ( F ) /\ B e. suc dom recs ( F ) ) -> ( ( C ` B ) = ( F ` ( C |` B ) ) <-> <. B , ( F ` ( C |` B ) ) >. e. C ) )
54	48, 52, 53	syl2anc 661	. . . . 5 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( ( C ` B ) = ( F ` ( C |` B ) ) <-> <. B , ( F ` ( C |` B ) ) >. e. C ) )
55	47, 54	mpbird 232	. . . 4 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C ` B ) = ( F ` ( C |` B ) ) )
56	55	ex 434	. . 3 |- ( dom recs ( F ) e. On -> ( B = dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
57	25, 56	jaod 380	. 2 |- ( dom recs ( F ) e. On -> ( ( B e. dom recs ( F ) \/ B = dom recs ( F ) ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
58	1, 57	syl5 32	1 |- ( dom recs ( F ) e. On -> ( B e. suc dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1374   e. wcel 1762   {cab 2445   A.wral 2807   E.wrex 2808   u. cun 3467   C_ wss 3469   {csn 4020   <.cop 4026   Oncon0 4871   suc csuc 4873   dom cdm 4992   |` cres 4994   Rel wrel 4997   Fun wfun 5573   Fn wfn 5574   ` cfv 5579  recscrecs 7031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587  df-recs 7032

This theorem is referenced by:  tfrlem12  7048