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Theorem dff13f 5765
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 5763 . 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 2319 . . . . . . . . 9  |-  F/_ y
w
42, 3nffv 5521 . . . . . . . 8  |-  F/_ y
( F `  w
)
5 nfcv 2319 . . . . . . . . 9  |-  F/_ y
v
62, 5nffv 5521 . . . . . . . 8  |-  F/_ y
( F `  v
)
74, 6nfeq 2327 . . . . . . 7  |-  F/ y ( F `  w
)  =  ( F `
 v )
8 nfv 1528 . . . . . . 7  |-  F/ y  w  =  v
97, 8nfim 1572 . . . . . 6  |-  F/ y ( ( F `  w )  =  ( F `  v )  ->  w  =  v )
10 nfv 1528 . . . . . 6  |-  F/ v ( ( F `  w )  =  ( F `  y )  ->  w  =  y )
11 fveq2 5511 . . . . . . . 8  |-  ( v  =  y  ->  ( F `  v )  =  ( F `  y ) )
1211eqeq2d 2189 . . . . . . 7  |-  ( v  =  y  ->  (
( F `  w
)  =  ( F `
 v )  <->  ( F `  w )  =  ( F `  y ) ) )
13 equequ2 1713 . . . . . . 7  |-  ( v  =  y  ->  (
w  =  v  <->  w  =  y ) )
1412, 13imbi12d 234 . . . . . 6  |-  ( v  =  y  ->  (
( ( F `  w )  =  ( F `  v )  ->  w  =  v )  <->  ( ( F `
 w )  =  ( F `  y
)  ->  w  =  y ) ) )
159, 10, 14cbvral 2699 . . . . 5  |-  ( A. v  e.  A  (
( F `  w
)  =  ( F `
 v )  ->  w  =  v )  <->  A. y  e.  A  ( ( F `  w
)  =  ( F `
 y )  ->  w  =  y )
)
1615ralbii 2483 . . . 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 2319 . . . . . 6  |-  F/_ x A
18 dff13f.1 . . . . . . . . 9  |-  F/_ x F
19 nfcv 2319 . . . . . . . . 9  |-  F/_ x w
2018, 19nffv 5521 . . . . . . . 8  |-  F/_ x
( F `  w
)
21 nfcv 2319 . . . . . . . . 9  |-  F/_ x
y
2218, 21nffv 5521 . . . . . . . 8  |-  F/_ x
( F `  y
)
2320, 22nfeq 2327 . . . . . . 7  |-  F/ x
( F `  w
)  =  ( F `
 y )
24 nfv 1528 . . . . . . 7  |-  F/ x  w  =  y
2523, 24nfim 1572 . . . . . 6  |-  F/ x
( ( F `  w )  =  ( F `  y )  ->  w  =  y )
2617, 25nfralxy 2515 . . . . 5  |-  F/ x A. y  e.  A  ( ( F `  w )  =  ( F `  y )  ->  w  =  y )
27 nfv 1528 . . . . 5  |-  F/ w A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y )
28 fveq2 5511 . . . . . . . 8  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
2928eqeq1d 2186 . . . . . . 7  |-  ( w  =  x  ->  (
( F `  w
)  =  ( F `
 y )  <->  ( F `  x )  =  ( F `  y ) ) )
30 equequ1 1712 . . . . . . 7  |-  ( w  =  x  ->  (
w  =  y  <->  x  =  y ) )
3129, 30imbi12d 234 . . . . . 6  |-  ( w  =  x  ->  (
( ( F `  w )  =  ( F `  y )  ->  w  =  y )  <->  ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) ) )
3231ralbidv 2477 . . . . 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 2699 . . . 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 184 . . 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 457 . 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 184 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 set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   F/_wnfc 2306   A.wral 2455   -->wf 5208   -1-1->wf1 5209   ` cfv 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fv 5220
This theorem is referenced by:  f1mpt  5766  dom2lem  6766
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