Theorem dff13f 5430

Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)

Hypotheses

Ref	Expression
dff13f.1	|- F/_ x F
dff13f.2	|- F/_ y F

Assertion

Ref	Expression
dff13f	|- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )

Distinct variable group:   x, y, A

Allowed substitution hints:   B( x, y)   F( x, y)

Proof of Theorem dff13f

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

Step	Hyp	Ref	Expression
1		dff13 5428	. 2 |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. w e. A A. v e. A ( ( F ` w ) = ( F ` v ) -> w = v ) ) )
2		dff13f.2	. . . . . . . . 9 |- F/_ y F
3		nfcv 2219	. . . . . . . . 9 |- F/_ y w
4	2, 3	nffv 5205	. . . . . . . 8 |- F/_ y ( F ` w )
5		nfcv 2219	. . . . . . . . 9 |- F/_ y v
6	2, 5	nffv 5205	. . . . . . . 8 |- F/_ y ( F ` v )
7	4, 6	nfeq 2226	. . . . . . 7 |- F/ y ( F ` w ) = ( F ` v )
8		nfv 1461	. . . . . . 7 |- F/ y w = v
9	7, 8	nfim 1504	. . . . . 6 |- F/ y ( ( F ` w ) = ( F ` v ) -> w = v )
10		nfv 1461	. . . . . 6 |- F/ v ( ( F ` w ) = ( F ` y ) -> w = y )
11		fveq2 5198	. . . . . . . 8 |- ( v = y -> ( F ` v ) = ( F ` y ) )
12	11	eqeq2d 2092	. . . . . . 7 |- ( v = y -> ( ( F ` w ) = ( F ` v ) <-> ( F ` w ) = ( F ` y ) ) )
13		equequ2 1639	. . . . . . 7 |- ( v = y -> ( w = v <-> w = y ) )
14	12, 13	imbi12d 232	. . . . . 6 |- ( v = y -> ( ( ( F ` w ) = ( F ` v ) -> w = v ) <-> ( ( F ` w ) = ( F ` y ) -> w = y ) ) )
15	9, 10, 14	cbvral 2573	. . . . 5 |- ( A. v e. A ( ( F ` w ) = ( F ` v ) -> w = v ) <-> A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y ) )
16	15	ralbii 2372	. . . 4 |- ( A. w e. A A. v e. A ( ( F ` w ) = ( F ` v ) -> w = v ) <-> A. w e. A A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y ) )
17		nfcv 2219	. . . . . 6 |- F/_ x A
18		dff13f.1	. . . . . . . . 9 |- F/_ x F
19		nfcv 2219	. . . . . . . . 9 |- F/_ x w
20	18, 19	nffv 5205	. . . . . . . 8 |- F/_ x ( F ` w )
21		nfcv 2219	. . . . . . . . 9 |- F/_ x y
22	18, 21	nffv 5205	. . . . . . . 8 |- F/_ x ( F ` y )
23	20, 22	nfeq 2226	. . . . . . 7 |- F/ x ( F ` w ) = ( F ` y )
24		nfv 1461	. . . . . . 7 |- F/ x w = y
25	23, 24	nfim 1504	. . . . . 6 |- F/ x ( ( F ` w ) = ( F ` y ) -> w = y )
26	17, 25	nfralxy 2402	. . . . 5 |- F/ x A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y )
27		nfv 1461	. . . . 5 |- F/ w A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y )
28		fveq2 5198	. . . . . . . 8 |- ( w = x -> ( F ` w ) = ( F ` x ) )
29	28	eqeq1d 2089	. . . . . . 7 |- ( w = x -> ( ( F ` w ) = ( F ` y ) <-> ( F ` x ) = ( F ` y ) ) )
30		equequ1 1638	. . . . . . 7 |- ( w = x -> ( w = y <-> x = y ) )
31	29, 30	imbi12d 232	. . . . . 6 |- ( w = x -> ( ( ( F ` w ) = ( F ` y ) -> w = y ) <-> ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
32	31	ralbidv 2368	. . . . 5 |- ( w = x -> ( A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y ) <-> A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
33	26, 27, 32	cbvral 2573	. . . 4 |- ( A. w e. A A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y ) <-> A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) )
34	16, 33	bitri 182	. . 3 |- ( A. w e. A A. v e. A ( ( F ` w ) = ( F ` v ) -> w = v ) <-> A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) )
35	34	anbi2i 444	. 2 |- ( ( F : A --> B /\ A. w e. A A. v e. A ( ( F ` w ) = ( F ` v ) -> w = v ) ) <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
36	1, 35	bitri 182	1 |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/_wnfc 2206   A.wral 2348   -->wf 4918   -1-1->wf1 4919   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fv 4930

This theorem is referenced by:  f1mpt  5431  dom2lem  6275