Theorem tfrlem11 6958

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 4894	. 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 6957	. . . . . . . 8 |- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) )
5		fnfun 5617	. . . . . . . 8 |- ( C Fn suc dom recs ( F ) -> Fun C )
6	4, 5	syl 16	. . . . . . 7 |- ( dom recs ( F ) e. On -> Fun C )
7		ssun1 3628	. . . . . . . . 9 |- recs ( F ) C_ (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
8	7, 3	sseqtr4i 3498	. . . . . . . 8 |- recs ( F ) C_ C
9	2	tfrlem9 6955	. . . . . . . . 9 |- ( B e. dom recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
10		funssfv 5815	. . . . . . . . . . . 12 |- ( ( Fun C /\ recs ( F ) C_ C /\ B e. dom recs ( F ) ) -> ( C ` B ) = (recs ( F ) ` B ) )
11	10	3expa 1188	. . . . . . . . . . 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 4870	. . . . . . . . . . . 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 5568	. . . . . . . . . . . . 13 |- ( ( Fun C /\ recs ( F ) C_ C /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
16	15	3expa 1188	. . . . . . . . . . . 12 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
17	16	fveq2d 5804	. . . . . . . . . . 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 2476	. . . . . . . . 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 4665	. . . . . . . . 9 |- <. B , ( F ` ( C |` B ) ) >. e. _V
27	26	snid 4014	. . . . . . . 8 |- <. B , ( F ` ( C |` B ) ) >. e. { <. B , ( F ` ( C |` B ) ) >. }
28		opeq1 4168	. . . . . . . . . . 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 3517	. . . . . . . . . . . . . 14 |- ( B = dom recs ( F ) -> B C_ dom recs ( F ) )
31	8, 15	mp3an2 1303	. . . . . . . . . . . . . 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 5214	. . . . . . . . . . . . . . 15 |- ( B = dom recs ( F ) -> (recs ( F ) |` B ) = (recs ( F ) |` dom recs ( F ) ) )
34	2	tfrlem6 6952	. . . . . . . . . . . . . . . 16 |- Rel recs ( F )
35		resdm 5257	. . . . . . . . . . . . . . . 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 2511	. . . . . . . . . . . . . 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 2495	. . . . . . . . . . . 12 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C |` B ) = recs ( F ) )
40	39	fveq2d 5804	. . . . . . . . . . 11 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( F ` ( C |` B ) ) = ( F ` recs ( F ) ) )
41	40	opeq2d 4175	. . . . . . . . . 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 2495	. . . . . . . . 9 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` recs ( F ) ) >. )
43	42	sneqd 3998	. . . . . . . 8 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> { <. B , ( F ` ( C |` B ) ) >. } = { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
44	27, 43	syl5eleq 2548	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
45		elun2 3633	. . . . . . 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 2553	. . . . 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 4906	. . . . . . . 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 2542	. . . . . 6 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> B e. suc dom recs ( F ) )
53		fnopfvb 5843	. . . . . 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 1370   e. wcel 1758   {cab 2439   A.wral 2799   E.wrex 2800   u. cun 3435   C_ wss 3437   {csn 3986   <.cop 3992   Oncon0 4828   suc csuc 4830   dom cdm 4949   |` cres 4951   Rel wrel 4954   Fun wfun 5521   Fn wfn 5522   ` cfv 5527  recscrecs 6942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-fv 5535  df-recs 6943

This theorem is referenced by:  tfrlem12  6959