Theorem tfri3 6638

Description: Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule G ( as described at tfri1 6636). 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 Jim Kingdon, 4-May-2019.)

Hypotheses

Ref	Expression
tfri3.1	|- F = recs ( G )
tfri3.2	|- ( Fun G /\ ( G ` x ) e. _V )

Assertion

Ref	Expression
tfri3	|- ( ( 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 tfri3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1581	. . . 4 |- F/ x B Fn On
2		nfra1 2581	. . . 4 |- F/ x A. x e. On ( B ` x ) = ( G ` ( B |` x ) )
3	1, 2	nfan 1618	. . 3 |- F/ x ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) )
4		nfv 1581	. . . . . 6 |- F/ x ( B ` y ) = ( F ` y )
5	3, 4	nfim 1625	. . . . 5 |- F/ x ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( B ` y ) = ( F ` y ) )
6		fveq2 5695	. . . . . . 7 |- ( x = y -> ( B ` x ) = ( B ` y ) )
7		fveq2 5695	. . . . . . 7 |- ( x = y -> ( F ` x ) = ( F ` y ) )
8	6, 7	eqeq12d 2253	. . . . . 6 |- ( x = y -> ( ( B ` x ) = ( F ` x ) <-> ( B ` y ) = ( F ` y ) ) )
9	8	imbi2d 230	. . . . 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 2627	. . . . . 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 2597	. . . . . . . . . 10 |- ( A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) -> ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) )
12		onss 4640	. . . . . . . . . . . . . . . . . . 19 |- ( x e. On -> x C_ On )
13		tfri3.1	. . . . . . . . . . . . . . . . . . . . . 22 |- F = recs ( G )
14		tfri3.2	. . . . . . . . . . . . . . . . . . . . . 22 |- ( Fun G /\ ( G ` x ) e. _V )
15	13, 14	tfri1 6636	. . . . . . . . . . . . . . . . . . . . 21 |- F Fn On
16		fvreseq 5812	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( B Fn On /\ F Fn On ) /\ x C_ On ) -> ( ( B |` x ) = ( F |` x ) <-> A. y e. x ( B ` y ) = ( F ` y ) ) )
17	15, 16	mpanl2 439	. . . . . . . . . . . . . . . . . . . 20 |- ( ( B Fn On /\ x C_ On ) -> ( ( B |` x ) = ( F |` x ) <-> A. y e. x ( B ` y ) = ( F ` y ) ) )
18		fveq2 5695	. . . . . . . . . . . . . . . . . . . 20 |- ( ( B |` x ) = ( F |` x ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) )
19	17, 18	biimtrrdi 164	. . . . . . . . . . . . . . . . . . 19 |- ( ( B Fn On /\ x C_ On ) -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
20	12, 19	sylan2 286	. . . . . . . . . . . . . . . . . 18 |- ( ( B Fn On /\ x e. On ) -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
21	20	ancoms 268	. . . . . . . . . . . . . . . . 17 |- ( ( x e. On /\ B Fn On ) -> ( A. y e. x ( B ` y ) = ( F ` y ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
22	21	imp 124	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. On /\ B Fn On ) /\ A. y e. x ( B ` y ) = ( F ` y ) ) -> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) )
23	22	adantr 276	. . . . . . . . . . . . . . 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 ) ) )
24	13, 14	tfri2 6637	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. On -> ( F ` x ) = ( G ` ( F |` x ) ) )
25	24	jctr 315	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
26		jcab 611	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
27	25, 26	sylibr 134	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( x e. On -> ( ( B ` x ) = ( G ` ( B |` x ) ) /\ ( F ` x ) = ( G ` ( F |` x ) ) ) ) )
28		eqeq12 2251	. . . . . . . . . . . . . . . . . 18 |- ( ( ( B ` x ) = ( G ` ( B |` x ) ) /\ ( F ` x ) = ( G ` ( F |` x ) ) ) -> ( ( B ` x ) = ( F ` x ) <-> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
29	27, 28	syl6 33	. . . . . . . . . . . . . . . . 17 |- ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( x e. On -> ( ( B ` x ) = ( F ` x ) <-> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) ) )
30	29	imp 124	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. On -> ( B ` x ) = ( G ` ( B |` x ) ) ) /\ x e. On ) -> ( ( B ` x ) = ( F ` x ) <-> ( G ` ( B |` x ) ) = ( G ` ( F |` x ) ) ) )
31	30	adantl 277	. . . . . . . . . . . . . . 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 ) ) ) )
32	23, 31	mpbird 167	. . . . . . . . . . . . . 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 ) )
33	32	exp43 372	. . . . . . . . . . . . 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 ) ) ) ) )
34	33	com4t 85	. . . . . . . . . . . 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 ) ) ) ) )
35	34	exp4a 366	. . . . . . . . . . 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 ) ) ) ) ) )
36	35	pm2.43d 50	. . . . . . . . . 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 ) ) ) ) )
37	11, 36	syl 14	. . . . . . . . 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 ) ) ) ) )
38	37	com3l 81	. . . . . . . 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 ) ) ) ) )
39	38	impd 254	. . . . . . 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 ) ) ) )
40	39	a2d 26	. . . . . 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 ) ) ) )
41	10, 40	biimtrid 152	. . . . 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 ) ) ) )
42	5, 9, 41	tfis2f 4731	. . . 4 |- ( x e. On -> ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( B ` x ) = ( F ` x ) ) )
43	42	com12 30	. . 3 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> ( x e. On -> ( B ` x ) = ( F ` x ) ) )
44	3, 43	ralrimi 2621	. 2 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> A. x e. On ( B ` x ) = ( F ` x ) )
45		eqfnfv 5806	. . . 4 |- ( ( B Fn On /\ F Fn On ) -> ( B = F <-> A. x e. On ( B ` x ) = ( F ` x ) ) )
46	15, 45	mpan2 429	. . 3 |- ( B Fn On -> ( B = F <-> A. x e. On ( B ` x ) = ( F ` x ) ) )
47	46	biimpar 297	. 2 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( F ` x ) ) -> B = F )
48	44, 47	syldan 282	1 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   Oncon0 4508   |` cres 4776   Fun wfun 5371   Fn wfn 5372   ` cfv 5377  recscrecs 6575

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-recs 6576

This theorem is used by: (None)