Theorem dff13f 5409

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 5407	. 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 2178	. . . . . . . . 9 |- F/_ y w
4	2, 3	nffv 5185	. . . . . . . 8 |- F/_ y ( F ` w )
5		nfcv 2178	. . . . . . . . 9 |- F/_ y v
6	2, 5	nffv 5185	. . . . . . . 8 |- F/_ y ( F ` v )
7	4, 6	nfeq 2185	. . . . . . 7 |- F/ y ( F ` w ) = ( F ` v )
8		nfv 1421	. . . . . . 7 |- F/ y w = v
9	7, 8	nfim 1464	. . . . . 6 |- F/ y ( ( F ` w ) = ( F ` v ) -> w = v )
10		nfv 1421	. . . . . 6 |- F/ v ( ( F ` w ) = ( F ` y ) -> w = y )
11		fveq2 5178	. . . . . . . 8 |- ( v = y -> ( F ` v ) = ( F ` y ) )
12	11	eqeq2d 2051	. . . . . . 7 |- ( v = y -> ( ( F ` w ) = ( F ` v ) <-> ( F ` w ) = ( F ` y ) ) )
13		equequ2 1599	. . . . . . 7 |- ( v = y -> ( w = v <-> w = y ) )
14	12, 13	imbi12d 223	. . . . . 6 |- ( v = y -> ( ( ( F ` w ) = ( F ` v ) -> w = v ) <-> ( ( F ` w ) = ( F ` y ) -> w = y ) ) )
15	9, 10, 14	cbvral 2529	. . . . 5 |- ( A. v e. A ( ( F ` w ) = ( F ` v ) -> w = v ) <-> A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y ) )
16	15	ralbii 2330	. . . 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 2178	. . . . . 6 |- F/_ x A
18		dff13f.1	. . . . . . . . 9 |- F/_ x F
19		nfcv 2178	. . . . . . . . 9 |- F/_ x w
20	18, 19	nffv 5185	. . . . . . . 8 |- F/_ x ( F ` w )
21		nfcv 2178	. . . . . . . . 9 |- F/_ x y
22	18, 21	nffv 5185	. . . . . . . 8 |- F/_ x ( F ` y )
23	20, 22	nfeq 2185	. . . . . . 7 |- F/ x ( F ` w ) = ( F ` y )
24		nfv 1421	. . . . . . 7 |- F/ x w = y
25	23, 24	nfim 1464	. . . . . 6 |- F/ x ( ( F ` w ) = ( F ` y ) -> w = y )
26	17, 25	nfralxy 2360	. . . . 5 |- F/ x A. y e. A ( ( F ` w ) = ( F ` y ) -> w = y )
27		nfv 1421	. . . . 5 |- F/ w A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y )
28		fveq2 5178	. . . . . . . 8 |- ( w = x -> ( F ` w ) = ( F ` x ) )
29	28	eqeq1d 2048	. . . . . . 7 |- ( w = x -> ( ( F ` w ) = ( F ` y ) <-> ( F ` x ) = ( F ` y ) ) )
30		equequ1 1598	. . . . . . 7 |- ( w = x -> ( w = y <-> x = y ) )
31	29, 30	imbi12d 223	. . . . . 6 |- ( w = x -> ( ( ( F ` w ) = ( F ` y ) -> w = y ) <-> ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
32	31	ralbidv 2326	. . . . 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 2529	. . . 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 173	. . 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 430	. 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 173	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 97   <-> wb 98   = wceq 1243   F/_wnfc 2165   A.wral 2306   -->wf 4898   -1-1->wf1 4899   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fv 4910

This theorem is referenced by:  f1mpt  5410  dom2lem  6252