Theorem tfr3 3917

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.

Hypotheses

Ref	Expression
tfr.1	|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
tfr.2	|- F = U.A

Assertion

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

Distinct variable groups:   x,y,f,A   x,F,y,f   x,G,y,f   x,B,y

Proof of Theorem tfr3

Step	Hyp	Ref	Expression
1		ax-17 969	. . . 4 |- (B Fn On -> A.x B Fn On)
2		hbra1 1684	. . . 4 |- (A.x e. On (B` x) = (G` (B |` x)) -> A.xA.x e. On (B` x) = (G` (B |` x)))
3	1, 2	hban 1007	. . 3 |- ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> A.x(B Fn On /\ A.x e. On (B` x) = (G` (B |` x))))
4		ax-17 969	. . . . . 6 |- ((B` y) = (F` y) -> A.x(B` y) = (F` y))
5	3, 4	hbim 1005	. . . . 5 |- (((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` y) = (F` y)) -> A.x((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` y) = (F` y)))
6		fveq2 3715	. . . . . . 7 |- (x = y -> (B` x) = (B` y))
7		fveq2 3715	. . . . . . 7 |- (x = y -> (F` x) = (F` y))
8	6, 7	eqeq12d 1486	. . . . . 6 |- (x = y -> ((B` x) = (F` x) <-> (B` y) = (F` y)))
9	8	imbi2d 611	. . . . 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		ra4 1691	. . . . . . . . . 10 |- (A.x e. On (B` x) = (G` (B |` x)) -> (x e. On -> (B` x) = (G` (B |` x))))
11		tfr.1	. . . . . . . . . . . . . . . . . . . . . 22 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
12		tfr.2	. . . . . . . . . . . . . . . . . . . . . 22 |- F = U.A
13	11, 12	tfr1 3915	. . . . . . . . . . . . . . . . . . . . 21 |- F Fn On
14		fvreseq 3790	. . . . . . . . . . . . . . . . . . . . 21 |- (((B Fn On /\ F Fn On) /\ x (_ On) -> ((B |` x) = (F |` x) <-> A.y e. x (B` y) = (F` y)))
15	13, 14	mpanl2 706	. . . . . . . . . . . . . . . . . . . 20 |- ((B Fn On /\ x (_ On) -> ((B |` x) = (F |` x) <-> A.y e. x (B` y) = (F` y)))
16		fveq2 3715	. . . . . . . . . . . . . . . . . . . 20 |- ((B |` x) = (F |` x) -> (G` (B |` x)) = (G` (F |` x)))
17	15, 16	syl6bir 215	. . . . . . . . . . . . . . . . . . 19 |- ((B Fn On /\ x (_ On) -> (A.y e. x (B` y) = (F` y) -> (G` (B |` x)) = (G` (F |` x))))
18		onsst 2987	. . . . . . . . . . . . . . . . . . 19 |- (x e. On -> x (_ On)
19	17, 18	sylan2 451	. . . . . . . . . . . . . . . . . 18 |- ((B Fn On /\ x e. On) -> (A.y e. x (B` y) = (F` y) -> (G` (B |` x)) = (G` (F |` x))))
20	19	ancoms 436	. . . . . . . . . . . . . . . . 17 |- ((x e. On /\ B Fn On) -> (A.y e. x (B` y) = (F` y) -> (G` (B |` x)) = (G` (F |` x))))
21	20	imp 350	. . . . . . . . . . . . . . . 16 |- (((x e. On /\ B Fn On) /\ A.y e. x (B` y) = (F` y)) -> (G` (B |` x)) = (G` (F |` x)))
22	21	adantr 389	. . . . . . . . . . . . . . 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	11, 12	tfr2 3916	. . . . . . . . . . . . . . . . . . . 20 |- (x e. On -> (F` x) = (G` (F |` x)))
24	23	jctr 291	. . . . . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . . . . . 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 200	. . . . . . . . . . . . . . . . . 18 |- ((x e. On -> (B` x) = (G` (B |` x))) -> (x e. On -> ((B` x) = (G` (B |` x)) /\ (F` x) = (G` (F |` x)))))
27		eqeq12 1484	. . . . . . . . . . . . . . . . . 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 22	. . . . . . . . . . . . . . . . 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 350	. . . . . . . . . . . . . . . 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 388	. . . . . . . . . . . . . . 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 196	. . . . . . . . . . . . . 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 384	. . . . . . . . . . . . 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 40	. . . . . . . . . . . 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 378	. . . . . . . . . . 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 65	. . . . . . . . . 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	10, 35	syl 10	. . . . . . . . 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 34	. . . . . . . 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	imp3a 361	. . . . . . 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 13	. . . . . 6 |- (x e. On -> (((B Fn On /\ A.x e. On (B` x) = (G` (B |` x