Theorem dff13f 5549

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 5547	. 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 2228	. . . . . . . . 9 |- F/_ y w
4	2, 3	nffv 5315	. . . . . . . 8 |- F/_ y ( F ` w )
5		nfcv 2228	. . . . . . . . 9 |- F/_ y v
6	2, 5	nffv 5315	. . . . . . . 8 |- F/_ y ( F ` v )
7	4, 6	nfeq 2236	. . . . . . 7 |- F/ y ( F ` w ) = ( F ` v )
8		nfv 1466	. . . . . . 7 |- F/ y w = v
9	7, 8	nfim 1509	. . . . . 6 |- F/ y ( ( F ` w ) = ( F ` v ) -> w = v )
10		nfv 1466	. . . . . 6 |- F/ v ( ( F ` w ) = ( F ` y ) -> w = y )
11		fveq2 5305	. . . . . . . 8 |- ( v = y -> ( F ` v ) = ( F ` y ) )
12	11	eqeq2d 2099	. . . . . . 7 |- ( v = y -> ( ( F ` w ) = ( F ` v ) <-> ( F ` w ) = ( F ` y ) ) )
13		equequ2 1646	. . . . . . 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 2586	. . . . 5 |- ( A. v e. A ( ( F ` w ) = ( F ` v ) -> w = v ) <-> A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y ) )
16	15	ralbii 2384	. . . 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 2228	. . . . . 6 |- F/_ x A
18		dff13f.1	. . . . . . . . 9 |- F/_ x F
19		nfcv 2228	. . . . . . . . 9 |- F/_ x w
20	18, 19	nffv 5315	. . . . . . . 8 |- F/_ x ( F ` w )
21		nfcv 2228	. . . . . . . . 9 |- F/_ x y
22	18, 21	nffv 5315	. . . . . . . 8 |- F/_ x ( F ` y )
23	20, 22	nfeq 2236	. . . . . . 7 |- F/ x ( F ` w ) = ( F ` y )
24		nfv 1466	. . . . . . 7 |- F/ x w = y
25	23, 24	nfim 1509	. . . . . 6 |- F/ x ( ( F ` w ) = ( F ` y ) -> w = y )
26	17, 25	nfralxy 2414	. . . . 5 |- F/ x A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y )
27		nfv 1466	. . . . 5 |- F/ w A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y )
28		fveq2 5305	. . . . . . . 8 |- ( w = x -> ( F ` w ) = ( F ` x ) )
29	28	eqeq1d 2096	. . . . . . 7 |- ( w = x -> ( ( F ` w ) = ( F ` y ) <-> ( F ` x ) = ( F ` y ) ) )
30		equequ1 1645	. . . . . . 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 2380	. . . . 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 2586	. . . 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 445	. 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 1289   F/_wnfc 2215   A.wral 2359   -->wf 5011   -1-1->wf1 5012   ` cfv 5015

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fv 5023

This theorem is referenced by:  f1mpt  5550  dom2lem  6489