Theorem tfr3 7495

Description: Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47. Finally, we show that F is unique. We do this by showing that any class B with the same properties of F that we showed in parts 1 and 2 is identical to F. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypothesis

Ref	Expression
tfr.1	|- F = recs ( G )

Assertion

Ref	Expression
tfr3	|- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F )

Distinct variable groups:   x, B   x, F   x, G

Proof of Theorem tfr3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 |- F/ x B Fn On
2		nfra1 2941	. . . 4 |- F/ x A. x e. On ( B ` x ) = ( G ` ( B |` x ) )
3	1, 2	nfan 1828	. . 3 |- F/ x ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) )
4		nfv 1843	. . . . . 6 |- F/ x ( B ` y ) = ( F ` y )
5	3, 4	nfim 1825	. . . . 5 |- F/ x ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( B ` y ) = ( F ` y ) )
6		fveq2 6191	. . . . . . 7 |- ( x = y -> ( B ` x ) = ( B ` y ) )
7		fveq2 6191	. . . . . . 7 |- ( x = y -> ( F ` x ) = ( F ` y ) )
8	6, 7	eqeq12d 2637	. . . . . 6 |- ( x = y -> ( ( B ` x ) = ( F ` x ) <-> ( B ` y ) = ( F ` y ) ) )
9	8	imbi2d 330	. . . . 5 |- ( x = y -> ( ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( B ` x ) = ( F ` x ) ) <-> ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( B ` y ) = ( F ` y ) ) ) )
10		r19.21v 2960	. . . . . 6 |- ( A. y e. x ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( B ` y ) = ( F ` y ) ) <-> ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> A. y e. x ( B ` y ) = ( F ` y ) ) )
11		rsp 2929	. . . . . . . . . 10 |- ( A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) -> ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) )
12		onss 6990	. . . . . . . . . . . . . . . . . . 19 |- ( x e. On -> x C_ On )
13		tfr.1	. . . . . . . . . . . . . . . . . . . . . 22 |- F = recs ( G )
14	13	tfr1 7493	. . . . . . . . . . . . . . . . . . . . 21 |- F Fn On
15		fvreseq 6319	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( B Fn On /\ F Fn On ) /\ x C_ On ) -> ( ( B |` x ) = ( F |` x ) <-> A. y e. x ( B ` y ) = ( F ` y ) ) )
16	14, 15	mpanl2 717	. . . . . . . . . . . . . . . . . . . 20 |- ( ( B Fn On /\ x C_ On ) -> ( ( B |` x ) = ( F |` x ) <-> A. y e. x ( B ` y ) = ( F ` y ) ) )
17		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( ( B |` x ) = ( F |` x ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) )
18	16, 17	syl6bir 244	. . . . . . . . . . . . . . . . . . 19 |- ( ( B Fn On /\ x C_ On ) -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
19	12, 18	sylan2 491	. . . . . . . . . . . . . . . . . 18 |- ( ( B Fn On /\ x e. On ) -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
20	19	ancoms 469	. . . . . . . . . . . . . . . . 17 |- ( ( x e. On /\ B Fn On ) -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
21	20	imp 445	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. On /\ B Fn On ) /\ A. y e. x ( B ` y ) = ( F ` y ) ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) )
22	21	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( x e. On /\ B Fn On ) /\ A. y e. x ( B ` y ) = ( F ` y ) ) /\ ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) /\ x e. On ) ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) )
23	13	tfr2 7494	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. On -> ( F ` x ) = ( G ` ( F |` x ) ) )
24	23	jctr 565	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) /\ ( x e. On -> ( F ` x ) = ( G ` ( F |` x ) ) ) ) )
25		jcab 907	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. On -> ( ( B ` x ) = ( G ` ( B |` x ) ) /\ ( F ` x ) = ( G ` ( F |` x ) ) ) ) <-> ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) /\ ( x e. On -> ( F ` x ) = ( G ` ( F |` x ) ) ) ) )
26	24, 25	sylibr 224	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( x e. On -> ( ( B ` x ) = ( G ` ( B |` x ) ) /\ ( F ` x ) = ( G ` ( F |` x ) ) ) ) )
27		eqeq12 2635	. . . . . . . . . . . . . . . . . 18 |- ( ( ( B ` x ) = ( G ` ( B |` x ) ) /\ ( F ` x ) = ( G ` ( F |` x ) ) ) -> ( ( B ` x ) = ( F ` x ) <-> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
28	26, 27	syl6 35	. . . . . . . . . . . . . . . . 17 |- ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( x e. On -> ( ( B ` x ) = ( F ` x ) <-> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) ) )
29	28	imp 445	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) /\ x e. On ) -> ( ( B ` x ) = ( F ` x ) <-> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
30	29	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( ( x e. On /\ B Fn On ) /\ A. y e. x ( B ` y ) = ( F ` y ) ) /\ ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) /\ x e. On ) ) -> ( ( B ` x ) = ( F ` x ) <-> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
31	22, 30	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ( ( x e. On /\ B Fn On ) /\ A. y e. x ( B ` y ) = ( F ` y ) ) /\ ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) /\ x e. On ) ) -> ( B ` x ) = ( F ` x ) )
32	31	exp43 640	. . . . . . . . . . . . 13 |- ( ( x e. On /\ B Fn On ) -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( x e. On -> ( B ` x ) = ( F ` x ) ) ) ) )
33	32	com4t 93	. . . . . . . . . . . 12 |- ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( x e. On -> ( ( x e. On /\ B Fn On ) -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( B ` x ) = ( F ` x ) ) ) ) )
34	33	exp4a 633	. . . . . . . . . . 11 |- ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( x e. On -> ( x e. On -> ( B Fn On -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( B ` x ) = ( F ` x ) ) ) ) ) )
35	34	pm2.43d 53	. . . . . . . . . 10 |- ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( x e. On -> ( B Fn On -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( B ` x ) = ( F ` x ) ) ) ) )
36	11, 35	syl 17	. . . . . . . . 9 |- ( A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) -> ( x e. On -> ( B Fn On -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( B ` x ) = ( F ` x ) ) ) ) )
37	36	com3l 89	. . . . . . . 8 |- ( x e. On -> ( B Fn On -> ( A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( B ` x ) = ( F ` x ) ) ) ) )
38	37	impd 447	. . . . . . 7 |- ( x e. On -> ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( B ` x ) = ( F ` x ) ) ) )
39	38	a2d 29	. . . . . 6 |- ( x e. On -> ( ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> A. y e. x ( B ` y ) = ( F ` y ) ) -> ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( B ` x ) = ( F ` x ) ) ) )
40	10, 39	syl5bi 232	. . . . 5 |- ( x e. On -> ( A. y e. x ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( B ` y ) = ( F ` y ) ) -> ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( B ` x ) = ( F ` x ) ) ) )
41	5, 9, 40	tfis2f 7055	. . . 4 |- ( x e. On -> ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( B ` x ) = ( F ` x ) ) )
42	41	com12 32	. . 3 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( x e. On -> ( B ` x ) = ( F ` x ) ) )
43	3, 42	ralrimi 2957	. 2 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> A. x e. On ( B ` x ) = ( F ` x ) )
44		eqfnfv 6311	. . . 4 |- ( ( B Fn On /\ F Fn On ) -> ( B = F <-> A. x e. On ( B ` x ) = ( F ` x ) ) )
45	14, 44	mpan2 707	. . 3 |- ( B Fn On -> ( B = F <-> A. x e. On ( B ` x ) = ( F ` x ) ) )
46	45	biimpar 502	. 2 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( F ` x ) ) -> B = F )
47	43, 46	syldan 487	1 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   |` cres 5116   Oncon0 5723   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-rep 4771  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-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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by: (None)