Theorem wfr3g 7413

Description: Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. (Contributed by Scott Fenton, 11-Feb-2011.)

Assertion

Ref	Expression
wfr3g	|- ( ( ( R We A /\ R Se A ) /\ ( F Fn A /\ A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) ) /\ ( G Fn A /\ A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> F = G )

Distinct variable groups:   y, A   y, F   y, G   y, H   y, R

Proof of Theorem wfr3g

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.26 3064	. . . . . . 7 |- ( A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) <-> ( A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) )
2		fveq2 6191	. . . . . . . . . . . 12 |- ( z = w -> ( F ` z ) = ( F ` w ) )
3		fveq2 6191	. . . . . . . . . . . 12 |- ( z = w -> ( G ` z ) = ( G ` w ) )
4	2, 3	eqeq12d 2637	. . . . . . . . . . 11 |- ( z = w -> ( ( F ` z ) = ( G ` z ) <-> ( F ` w ) = ( G ` w ) ) )
5	4	imbi2d 330	. . . . . . . . . 10 |- ( z = w -> ( ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( F ` z ) = ( G ` z ) ) <-> ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( F ` w ) = ( G ` w ) ) ) )
6		ra4v 3524	. . . . . . . . . . 11 |- ( A. w e. Pred ( R , A , z ) ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( F ` w ) = ( G ` w ) ) -> ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) ) )
7		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 |- ( y = z -> ( F ` y ) = ( F ` z ) )
8		predeq3 5684	. . . . . . . . . . . . . . . . . . . . 21 |- ( y = z -> Pred ( R , A , y ) = Pred ( R , A , z ) )
9	8	reseq2d 5396	. . . . . . . . . . . . . . . . . . . 20 |- ( y = z -> ( F |` Pred ( R , A , y ) ) = ( F |` Pred ( R , A , z ) ) )
10	9	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( y = z -> ( H ` ( F |` Pred ( R , A , y ) ) ) = ( H ` ( F |` Pred ( R , A , z ) ) ) )
11	7, 10	eqeq12d 2637	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) <-> ( F ` z ) = ( H ` ( F |` Pred ( R , A , z ) ) ) ) )
12		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 |- ( y = z -> ( G ` y ) = ( G ` z ) )
13	8	reseq2d 5396	. . . . . . . . . . . . . . . . . . . 20 |- ( y = z -> ( G |` Pred ( R , A , y ) ) = ( G |` Pred ( R , A , z ) ) )
14	13	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( y = z -> ( H ` ( G |` Pred ( R , A , y ) ) ) = ( H ` ( G |` Pred ( R , A , z ) ) ) )
15	12, 14	eqeq12d 2637	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) <-> ( G ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) ) )
16	11, 15	anbi12d 747	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) <-> ( ( F ` z ) = ( H ` ( F |` Pred ( R , A , z ) ) ) /\ ( G ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) ) ) )
17	16	rspcva 3307	. . . . . . . . . . . . . . . 16 |- ( ( z e. A /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( ( F ` z ) = ( H ` ( F |` Pred ( R , A , z ) ) ) /\ ( G ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) ) )
18		eqtr3 2643	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( F ` z ) = ( H ` ( F |` Pred ( R , A , z ) ) ) /\ ( H ` ( G |` Pred ( R , A , z ) ) ) = ( H ` ( F |` Pred ( R , A , z ) ) ) ) -> ( F ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) )
19	18	ancoms 469	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( H ` ( G |` Pred ( R , A , z ) ) ) = ( H ` ( F |` Pred ( R , A , z ) ) ) /\ ( F ` z ) = ( H ` ( F |` Pred ( R , A , z ) ) ) ) -> ( F ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) )
20		eqtr3 2643	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( F ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) /\ ( G ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) ) -> ( F ` z ) = ( G ` z ) )
21	20	ex 450	. . . . . . . . . . . . . . . . . . . 20 |- ( ( F ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) -> ( ( G ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) -> ( F ` z ) = ( G ` z ) ) )
22	19, 21	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( H ` ( G |` Pred ( R , A , z ) ) ) = ( H ` ( F |` Pred ( R , A , z ) ) ) /\ ( F ` z ) = ( H ` ( F |` Pred ( R , A , z ) ) ) ) -> ( ( G ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) -> ( F ` z ) = ( G ` z ) ) )
23	22	expimpd 629	. . . . . . . . . . . . . . . . . 18 |- ( ( H ` ( G |` Pred ( R , A , z ) ) ) = ( H ` ( F |` Pred ( R , A , z ) ) ) -> ( ( ( F ` z ) = ( H ` ( F |` Pred ( R , A , z ) ) ) /\ ( G ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) ) -> ( F ` z ) = ( G ` z ) ) )
24		predss 5687	. . . . . . . . . . . . . . . . . . . . . 22 |- Pred ( R , A , z ) C_ A
25		fvreseq 6319	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( F Fn A /\ G Fn A ) /\ Pred ( R , A , z ) C_ A ) -> ( ( F |` Pred ( R , A , z ) ) = ( G |` Pred ( R , A , z ) ) <-> A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) ) )
26	24, 25	mpan2 707	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( F Fn A /\ G Fn A ) -> ( ( F |` Pred ( R , A , z ) ) = ( G |` Pred ( R , A , z ) ) <-> A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) ) )
27	26	biimpar 502	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( F Fn A /\ G Fn A ) /\ A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) ) -> ( F |` Pred ( R , A , z ) ) = ( G |` Pred ( R , A , z ) ) )
28	27	eqcomd 2628	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F Fn A /\ G Fn A ) /\ A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) ) -> ( G |` Pred ( R , A , z ) ) = ( F |` Pred ( R , A , z ) ) )
29	28	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F Fn A /\ G Fn A ) /\ A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) ) -> ( H ` ( G |` Pred ( R , A , z ) ) ) = ( H ` ( F |` Pred ( R , A , z ) ) ) )
30	23, 29	syl11 33	. . . . . . . . . . . . . . . . 17 |- ( ( ( F ` z ) = ( H ` ( F |` Pred ( R , A , z ) ) ) /\ ( G ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) ) -> ( ( ( F Fn A /\ G Fn A ) /\ A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) ) -> ( F ` z ) = ( G ` z ) ) )
31	30	expd 452	. . . . . . . . . . . . . . . 16 |- ( ( ( F ` z ) = ( H ` ( F |` Pred ( R , A , z ) ) ) /\ ( G ` z ) = ( H ` ( G |` Pred ( R , A , z ) ) ) ) -> ( ( F Fn A /\ G Fn A ) -> ( A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) -> ( F ` z ) = ( G ` z ) ) ) )
32	17, 31	syl 17	. . . . . . . . . . . . . . 15 |- ( ( z e. A /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( ( F Fn A /\ G Fn A ) -> ( A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) -> ( F ` z ) = ( G ` z ) ) ) )
33	32	ex 450	. . . . . . . . . . . . . 14 |- ( z e. A -> ( A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) -> ( ( F Fn A /\ G Fn A ) -> ( A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) -> ( F ` z ) = ( G ` z ) ) ) ) )
34	33	com23 86	. . . . . . . . . . . . 13 |- ( z e. A -> ( ( F Fn A /\ G Fn A ) -> ( A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) -> ( A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) -> ( F ` z ) = ( G ` z ) ) ) ) )
35	34	impd 447	. . . . . . . . . . . 12 |- ( z e. A -> ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) -> ( F ` z ) = ( G ` z ) ) ) )
36	35	a2d 29	. . . . . . . . . . 11 |- ( z e. A -> ( ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> A. w e. Pred ( R , A , z ) ( F ` w ) = ( G ` w ) ) -> ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( F ` z ) = ( G ` z ) ) ) )
37	6, 36	syl5 34	. . . . . . . . . 10 |- ( z e. A -> ( A. w e. Pred ( R , A , z ) ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( F ` w ) = ( G ` w ) ) -> ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( F ` z ) = ( G ` z ) ) ) )
38	5, 37	wfis2g 5719	. . . . . . . . 9 |- ( ( R We A /\ R Se A ) -> A. z e. A ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( F ` z ) = ( G ` z ) ) )
39		r19.21v 2960	. . . . . . . . 9 |- ( A. z e. A ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( F ` z ) = ( G ` z ) ) <-> ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> A. z e. A ( F ` z ) = ( G ` z ) ) )
40	38, 39	sylib 208	. . . . . . . 8 |- ( ( R We A /\ R Se A ) -> ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> A. z e. A ( F ` z ) = ( G ` z ) ) )
41	40	com12 32	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn A ) /\ A. y e. A ( ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( ( R We A /\ R Se A ) -> A. z e. A ( F ` z ) = ( G ` z ) ) )
42	1, 41	sylan2br 493	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ ( A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) /\ A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( ( R We A /\ R Se A ) -> A. z e. A ( F ` z ) = ( G ` z ) ) )
43	42	an4s 869	. . . . 5 |- ( ( ( F Fn A /\ A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) ) /\ ( G Fn A /\ A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( ( R We A /\ R Se A ) -> A. z e. A ( F ` z ) = ( G ` z ) ) )
44	43	com12 32	. . . 4 |- ( ( R We A /\ R Se A ) -> ( ( ( F Fn A /\ A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) ) /\ ( G Fn A /\ A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> A. z e. A ( F ` z ) = ( G ` z ) ) )
45	44	3impib 1262	. . 3 |- ( ( ( R We A /\ R Se A ) /\ ( F Fn A /\ A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) ) /\ ( G Fn A /\ A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> A. z e. A ( F ` z ) = ( G ` z ) )
46		eqid 2622	. . 3 |- A = A
47	45, 46	jctil 560	. 2 |- ( ( ( R We A /\ R Se A ) /\ ( F Fn A /\ A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) ) /\ ( G Fn A /\ A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( A = A /\ A. z e. A ( F ` z ) = ( G ` z ) ) )
48		eqfnfv2 6312	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> ( A = A /\ A. z e. A ( F ` z ) = ( G ` z ) ) ) )
49	48	ad2ant2r 783	. . 3 |- ( ( ( F Fn A /\ A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) ) /\ ( G Fn A /\ A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( F = G <-> ( A = A /\ A. z e. A ( F ` z ) = ( G ` z ) ) ) )
50	49	3adant1 1079	. 2 |- ( ( ( R We A /\ R Se A ) /\ ( F Fn A /\ A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) ) /\ ( G Fn A /\ A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> ( F = G <-> ( A = A /\ A. z e. A ( F ` z ) = ( G ` z ) ) ) )
51	47, 50	mpbird 247	1 |- ( ( ( R We A /\ R Se A ) /\ ( F Fn A /\ A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) ) /\ ( G Fn A /\ A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) ) -> F = G )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   Se wse 5071   We wwe 5072   |` cres 5116   Predcpred 5679   Fn wfn 5883   ` cfv 5888

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

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-rmo 2920  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  wfrlem5  7419  wfr3  7435