Theorem tfrlem11 7484

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 5791	. 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 7483	. . . . . . . 8 |- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) )
5		fnfun 5988	. . . . . . . 8 |- ( C Fn suc dom recs ( F ) -> Fun C )
6	4, 5	syl 17	. . . . . . 7 |- ( dom recs ( F ) e. On -> Fun C )
7		ssun1 3776	. . . . . . . . 9 |- recs ( F ) C_ (recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
8	7, 3	sseqtr4i 3638	. . . . . . . 8 |- recs ( F ) C_ C
9	2	tfrlem9 7481	. . . . . . . . 9 |- ( B e. dom recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
10		funssfv 6209	. . . . . . . . . . . 12 |- ( ( Fun C /\ recs ( F ) C_ C /\ B e. dom recs ( F ) ) -> ( C ` B ) = (recs ( F ) ` B ) )
11	10	3expa 1265	. . . . . . . . . . 11 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B e. dom recs ( F ) ) -> ( C ` B ) = (recs ( F ) ` B ) )
12	11	adantrl 752	. . . . . . . . . 10 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( C ` B ) = (recs ( F ) ` B ) )
13		onelss 5766	. . . . . . . . . . . 12 |- ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> B C_ dom recs ( F ) ) )
14	13	imp 445	. . . . . . . . . . 11 |- ( ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) -> B C_ dom recs ( F ) )
15		fun2ssres 5931	. . . . . . . . . . . . 13 |- ( ( Fun C /\ recs ( F ) C_ C /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
16	15	3expa 1265	. . . . . . . . . . . 12 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
17	16	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B C_ dom recs ( F ) ) -> ( F ` ( C |` B ) ) = ( F ` (recs ( F ) |` B ) ) )
18	14, 17	sylan2 491	. . . . . . . . . 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 2637	. . . . . . . . 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 236	. . . . . . . 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 717	. . . . . . 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 488	. . . . . 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 631	. . . . 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 52	. . . 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 53	. . 3 |- ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
26		opex 4932	. . . . . . . . 9 |- <. B , ( F ` ( C |` B ) ) >. e. _V
27	26	snid 4208	. . . . . . . 8 |- <. B , ( F ` ( C |` B ) ) >. e. { <. B , ( F ` ( C |` B ) ) >. }
28		opeq1 4402	. . . . . . . . . . 11 |- ( B = dom recs ( F ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` ( C |` B ) ) >. )
29	28	adantl 482	. . . . . . . . . 10 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` ( C |` B ) ) >. )
30		eqimss 3657	. . . . . . . . . . . . . 14 |- ( B = dom recs ( F ) -> B C_ dom recs ( F ) )
31	8, 15	mp3an2 1412	. . . . . . . . . . . . . 14 |- ( ( Fun C /\ B C_ dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
32	6, 30, 31	syl2an 494	. . . . . . . . . . . . 13 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C |` B ) = (recs ( F ) |` B ) )
33		reseq2 5391	. . . . . . . . . . . . . . 15 |- ( B = dom recs ( F ) -> (recs ( F ) |` B ) = (recs ( F ) |` dom recs ( F ) ) )
34	2	tfrlem6 7478	. . . . . . . . . . . . . . . 16 |- Rel recs ( F )
35		resdm 5441	. . . . . . . . . . . . . . . 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 2672	. . . . . . . . . . . . . 14 |- ( B = dom recs ( F ) -> (recs ( F ) |` B ) = recs ( F ) )
38	37	adantl 482	. . . . . . . . . . . . 13 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> (recs ( F ) |` B ) = recs ( F ) )
39	32, 38	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C |` B ) = recs ( F ) )
40	39	fveq2d 6195	. . . . . . . . . . 11 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( F ` ( C |` B ) ) = ( F ` recs ( F ) ) )
41	40	opeq2d 4409	. . . . . . . . . 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 2656	. . . . . . . . 9 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` recs ( F ) ) >. )
43	42	sneqd 4189	. . . . . . . 8 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> { <. B , ( F ` ( C |` B ) ) >. } = { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
44	27, 43	syl5eleq 2707	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } )
45		elun2 3781	. . . . . . 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 2712	. . . . 5 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. C )
48	4	adantr 481	. . . . . 6 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> C Fn suc dom recs ( F ) )
49		simpr 477	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> B = dom recs ( F ) )
50		sucidg 5803	. . . . . . . 8 |- ( dom recs ( F ) e. On -> dom recs ( F ) e. suc dom recs ( F ) )
51	50	adantr 481	. . . . . . 7 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> dom recs ( F ) e. suc dom recs ( F ) )
52	49, 51	eqeltrd 2701	. . . . . 6 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> B e. suc dom recs ( F ) )
53		fnopfvb 6237	. . . . . 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 693	. . . . 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 247	. . . 4 |- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C ` B ) = ( F ` ( C |` B ) ) )
56	55	ex 450	. . 3 |- ( dom recs ( F ) e. On -> ( B = dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
57	25, 56	jaod 395	. 2 |- ( dom recs ( F ) e. On -> ( ( B e. dom recs ( F ) \/ B = dom recs ( F ) ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )
58	1, 57	syl5 34	1 |- ( dom recs ( F ) e. On -> ( B e. suc dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   u. cun 3572   C_ wss 3574   {csn 4177   <.cop 4183   dom cdm 5114   |` cres 5116   Rel wrel 5119   Oncon0 5723   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  recscrecs 7467

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-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-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-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  tfrlem12  7485