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Theorem dff13f 5971
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.
StepHypRef Expression
1 dff13 5969 . 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 2577 . . . . . . . . 9  |-  F/_ y
w
42, 3nffv 5696 . . . . . . . 8  |-  F/_ y
( F `  w
)
5 nfcv 2577 . . . . . . . . 9  |-  F/_ y
v
62, 5nffv 5696 . . . . . . . 8  |-  F/_ y
( F `  v
)
74, 6nfeq 2584 . . . . . . 7  |-  F/ y ( F `  w
)  =  ( F `
 v )
8 nfv 1673 . . . . . . 7  |-  F/ y  w  =  v
97, 8nfim 1853 . . . . . 6  |-  F/ y ( ( F `  w )  =  ( F `  v )  ->  w  =  v )
10 nfv 1673 . . . . . 6  |-  F/ v ( ( F `  w )  =  ( F `  y )  ->  w  =  y )
11 fveq2 5689 . . . . . . . 8  |-  ( v  =  y  ->  ( F `  v )  =  ( F `  y ) )
1211eqeq2d 2452 . . . . . . 7  |-  ( v  =  y  ->  (
( F `  w
)  =  ( F `
 v )  <->  ( F `  w )  =  ( F `  y ) ) )
13 equequ2 1737 . . . . . . 7  |-  ( v  =  y  ->  (
w  =  v  <->  w  =  y ) )
1412, 13imbi12d 320 . . . . . 6  |-  ( v  =  y  ->  (
( ( F `  w )  =  ( F `  v )  ->  w  =  v )  <->  ( ( F `
 w )  =  ( F `  y
)  ->  w  =  y ) ) )
159, 10, 14cbvral 2941 . . . . 5  |-  ( A. v  e.  A  (
( F `  w
)  =  ( F `
 v )  ->  w  =  v )  <->  A. y  e.  A  ( ( F `  w
)  =  ( F `
 y )  ->  w  =  y )
)
1615ralbii 2737 . . . 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 2577 . . . . . 6  |-  F/_ x A
18 dff13f.1 . . . . . . . . 9  |-  F/_ x F
19 nfcv 2577 . . . . . . . . 9  |-  F/_ x w
2018, 19nffv 5696 . . . . . . . 8  |-  F/_ x
( F `  w
)
21 nfcv 2577 . . . . . . . . 9  |-  F/_ x
y
2218, 21nffv 5696 . . . . . . . 8  |-  F/_ x
( F `  y
)
2320, 22nfeq 2584 . . . . . . 7  |-  F/ x
( F `  w
)  =  ( F `
 y )
24 nfv 1673 . . . . . . 7  |-  F/ x  w  =  y
2523, 24nfim 1853 . . . . . 6  |-  F/ x
( ( F `  w )  =  ( F `  y )  ->  w  =  y )
2617, 25nfral 2767 . . . . 5  |-  F/ x A. y  e.  A  ( ( F `  w )  =  ( F `  y )  ->  w  =  y )
27 nfv 1673 . . . . 5  |-  F/ w A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y )
28 fveq2 5689 . . . . . . . 8  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
2928eqeq1d 2449 . . . . . . 7  |-  ( w  =  x  ->  (
( F `  w
)  =  ( F `
 y )  <->  ( F `  x )  =  ( F `  y ) ) )
30 equequ1 1736 . . . . . . 7  |-  ( w  =  x  ->  (
w  =  y  <->  x  =  y ) )
3129, 30imbi12d 320 . . . . . 6  |-  ( w  =  x  ->  (
( ( F `  w )  =  ( F `  y )  ->  w  =  y )  <->  ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) ) )
3231ralbidv 2733 . . . . 5  |-  ( w  =  x  ->  ( A. y  e.  A  ( ( F `  w )  =  ( F `  y )  ->  w  =  y )  <->  A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
3326, 27, 32cbvral 2941 . . . 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 )
)
3416, 33bitri 249 . . 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 )
)
3534anbi2i 694 . 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 ) ) )
361, 35bitri 249 1  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   F/_wnfc 2564   A.wral 2713   -->wf 5412   -1-1->wf1 5413   ` cfv 5416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fv 5424
This theorem is referenced by:  f1mpt  5972  dom2lem  7347
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