Theorem tfrlem11 7105

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 5499	. 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 7104	. . . . . . . 8 |- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) )
5		fnfun 5682	. . . . . . . 8 |- ( C Fn suc dom recs ( F ) -> Fun C )
6	4, 5	syl 17	. . . . . . 7 |- ( dom recs ( F ) e. On -> Fun C )
7		ssun1 3626	. . . . . . . . 9 |- recs ( F ) C_ (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
8	7, 3	sseqtr4i 3494	. . . . . . . 8 |- recs ( F ) C_ C
9	2	tfrlem9 7102	. . . . . . . . 9 |- ( B e. dom recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
10		funssfv 5887	. . . . . . . . . . . 12 |- ( ( Fun C /\ recs ( F ) C_ C /\ B e. dom recs ( F ) ) -> ( C ` B ) = (recs ( F ) ` B ) )
11	10	3expa 1205	. . . . . . . . . . 11 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B e. dom recs ( F ) ) -> ( C ` B ) = (recs ( F ) ` B ) )
12	11	adantrl 720	. . . . . . . . . 10 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( C ` B ) = (recs ( F ) ` B ) )
13		onelss 5475	. . . . . . . . . . . 12 |- ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> B C_ dom recs ( F ) ) )
14	13	imp 430	. . . . . . . . . . 11 |- ( ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) -> B C_ dom recs ( F ) )
15		fun2ssres 5633	. . . . . . . . . . . . 13 |- ( ( Fun C /\ recs ( F ) C_ C /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
16	15	3expa 1205	. . . . . . . . . . . 12 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
17	16	fveq2d 5876	. . . . . . . . . . 11 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B C_ dom recs ( F ) ) -> ( F ` ( C |` B ) ) = ( F ` (recs ( F ) |` B ) ) )
18	14, 17	sylan2 476	. . . . . . . . . 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 2442	. . . . . . . . 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 224	. . . . . . . 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 685	. . . . . . 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 473	. . . . . 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 608	. . . . 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 49	. . . 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 50	. . 3 |- ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
26		opex 4677	. . . . . . . . 9 |- <. B , ( F ` ( C |` B ) ) >. e. _V
27	26	snid 4021	. . . . . . . 8 |- <. B , ( F ` ( C |` B ) ) >. e. { <. B , ( F ` ( C |` B ) ) >. }
28		opeq1 4181	. . . . . . . . . . 11 |- ( B = dom recs ( F ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` ( C |` B ) ) >. )
29	28	adantl 467	. . . . . . . . . 10 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` ( C |` B ) ) >. )
30		eqimss 3513	. . . . . . . . . . . . . 14 |- ( B = dom recs ( F ) -> B C_ dom recs ( F ) )
31	8, 15	mp3an2 1348	. . . . . . . . . . . . . 14 |- ( ( Fun C /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
32	6, 30, 31	syl2an 479	. . . . . . . . . . . . 13 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
33		reseq2 5111	. . . . . . . . . . . . . . 15 |- ( B = dom recs ( F ) -> (recs ( F ) |` B ) = (recs ( F ) |` dom recs ( F ) ) )
34	2	tfrlem6 7099	. . . . . . . . . . . . . . . 16 |- Rel recs ( F )
35		resdm 5157	. . . . . . . . . . . . . . . 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 2477	. . . . . . . . . . . . . 14 |- ( B = dom recs ( F ) -> (recs ( F ) |` B ) = recs ( F ) )
38	37	adantl 467	. . . . . . . . . . . . 13 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> (recs ( F ) |` B ) = recs ( F ) )
39	32, 38	eqtrd 2461	. . . . . . . . . . . 12 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C |` B ) = recs ( F ) )
40	39	fveq2d 5876	. . . . . . . . . . 11 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( F ` ( C |` B ) ) = ( F ` recs ( F ) ) )
41	40	opeq2d 4188	. . . . . . . . . 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 2461	. . . . . . . . 9 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` recs ( F ) ) >. )
43	42	sneqd 4005	. . . . . . . 8 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> { <. B , ( F ` ( C |` B ) ) >. } = { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
44	27, 43	syl5eleq 2514	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
45		elun2 3631	. . . . . . 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 17	. . . . . 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 2519	. . . . 5 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. C )
48	4	adantr 466	. . . . . 6 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> C Fn suc dom recs ( F ) )
49		simpr 462	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> B = dom recs ( F ) )
50		sucidg 5511	. . . . . . . 8 |- ( dom recs ( F ) e. On -> dom recs ( F ) e. suc dom recs ( F ) )
51	50	adantr 466	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> dom recs ( F ) e. suc dom recs ( F ) )
52	49, 51	eqeltrd 2508	. . . . . 6 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> B e. suc dom recs ( F ) )
53		fnopfvb 5913	. . . . . 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 665	. . . . 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 235	. . . 4 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C ` B ) = ( F ` ( C |` B ) ) )
56	55	ex 435	. . 3 |- ( dom recs ( F ) e. On -> ( B = dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
57	25, 56	jaod 381	. 2 |- ( dom recs ( F ) e. On -> ( ( B e. dom recs ( F ) \/ B = dom recs ( F ) ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
58	1, 57	syl5 33	1 |- ( dom recs ( F ) e. On -> ( B e. suc dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   = wceq 1437   e. wcel 1867   {cab 2405   A.wral 2773   E.wrex 2774   u. cun 3431   C_ wss 3433   {csn 3993   <.cop 3999   dom cdm 4845   |` cres 4847   Rel wrel 4850   Oncon0 5433   suc csuc 5435   Fun wfun 5586   Fn wfn 5587   ` cfv 5592  recscrecs 7088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588

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 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-fv 5600  df-wrecs 7027  df-recs 7089

This theorem is referenced by:  tfrlem12  7106