Theorem dff13f 5749

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 5747	. 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 2312	. . . . . . . . 9 |- F/_ y w
4	2, 3	nffv 5506	. . . . . . . 8 |- F/_ y ( F ` w )
5		nfcv 2312	. . . . . . . . 9 |- F/_ y v
6	2, 5	nffv 5506	. . . . . . . 8 |- F/_ y ( F ` v )
7	4, 6	nfeq 2320	. . . . . . 7 |- F/ y ( F ` w ) = ( F ` v )
8		nfv 1521	. . . . . . 7 |- F/ y w = v
9	7, 8	nfim 1565	. . . . . 6 |- F/ y ( ( F ` w ) = ( F ` v ) -> w = v )
10		nfv 1521	. . . . . 6 |- F/ v ( ( F ` w ) = ( F ` y ) -> w = y )
11		fveq2 5496	. . . . . . . 8 |- ( v = y -> ( F ` v ) = ( F ` y ) )
12	11	eqeq2d 2182	. . . . . . 7 |- ( v = y -> ( ( F ` w ) = ( F ` v ) <-> ( F ` w ) = ( F ` y ) ) )
13		equequ2 1706	. . . . . . 7 |- ( v = y -> ( w = v <-> w = y ) )
14	12, 13	imbi12d 233	. . . . . 6 |- ( v = y -> ( ( ( F ` w ) = ( F ` v ) -> w = v ) <-> ( ( F ` w ) = ( F ` y ) -> w = y ) ) )
15	9, 10, 14	cbvral 2692	. . . . 5 |- ( A. v e. A ( ( F ` w ) = ( F ` v ) -> w = v ) <-> A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y ) )
16	15	ralbii 2476	. . . 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 2312	. . . . . 6 |- F/_ x A
18		dff13f.1	. . . . . . . . 9 |- F/_ x F
19		nfcv 2312	. . . . . . . . 9 |- F/_ x w
20	18, 19	nffv 5506	. . . . . . . 8 |- F/_ x ( F ` w )
21		nfcv 2312	. . . . . . . . 9 |- F/_ x y
22	18, 21	nffv 5506	. . . . . . . 8 |- F/_ x ( F ` y )
23	20, 22	nfeq 2320	. . . . . . 7 |- F/ x ( F ` w ) = ( F ` y )
24		nfv 1521	. . . . . . 7 |- F/ x w = y
25	23, 24	nfim 1565	. . . . . 6 |- F/ x ( ( F ` w ) = ( F ` y ) -> w = y )
26	17, 25	nfralxy 2508	. . . . 5 |- F/ x A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y )
27		nfv 1521	. . . . 5 |- F/ w A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y )
28		fveq2 5496	. . . . . . . 8 |- ( w = x -> ( F ` w ) = ( F ` x ) )
29	28	eqeq1d 2179	. . . . . . 7 |- ( w = x -> ( ( F ` w ) = ( F ` y ) <-> ( F ` x ) = ( F ` y ) ) )
30		equequ1 1705	. . . . . . 7 |- ( w = x -> ( w = y <-> x = y ) )
31	29, 30	imbi12d 233	. . . . . 6 |- ( w = x -> ( ( ( F ` w ) = ( F ` y ) -> w = y ) <-> ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
32	31	ralbidv 2470	. . . . 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 2692	. . . 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 183	. . 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 454	. 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 183	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 103   <-> wb 104   = wceq 1348   F/_wnfc 2299   A.wral 2448   -->wf 5194   -1-1->wf1 5195   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fv 5206

This theorem is referenced by:  f1mpt  5750  dom2lem  6750