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Theorem dff13f 5786
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 5785 . 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 2421 . . . . . . . . 9  |-  F/_ y
w
42, 3nffv 5534 . . . . . . . 8  |-  F/_ y
( F `  w
)
5 nfcv 2421 . . . . . . . . 9  |-  F/_ y
v
62, 5nffv 5534 . . . . . . . 8  |-  F/_ y
( F `  v
)
74, 6nfeq 2428 . . . . . . 7  |-  F/ y ( F `  w
)  =  ( F `
 v )
8 nfv 1607 . . . . . . 7  |-  F/ y  w  =  v
97, 8nfim 1771 . . . . . 6  |-  F/ y ( ( F `  w )  =  ( F `  v )  ->  w  =  v )
10 nfv 1607 . . . . . 6  |-  F/ v ( ( F `  w )  =  ( F `  y )  ->  w  =  y )
11 fveq2 5527 . . . . . . . 8  |-  ( v  =  y  ->  ( F `  v )  =  ( F `  y ) )
1211eqeq2d 2296 . . . . . . 7  |-  ( v  =  y  ->  (
( F `  w
)  =  ( F `
 v )  <->  ( F `  w )  =  ( F `  y ) ) )
13 eqeq2 2294 . . . . . . 7  |-  ( v  =  y  ->  (
w  =  v  <->  w  =  y ) )
1412, 13imbi12d 311 . . . . . 6  |-  ( v  =  y  ->  (
( ( F `  w )  =  ( F `  v )  ->  w  =  v )  <->  ( ( F `
 w )  =  ( F `  y
)  ->  w  =  y ) ) )
159, 10, 14cbvral 2762 . . . . 5  |-  ( A. v  e.  A  (
( F `  w
)  =  ( F `
 v )  ->  w  =  v )  <->  A. y  e.  A  ( ( F `  w
)  =  ( F `
 y )  ->  w  =  y )
)
1615ralbii 2569 . . . 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 2421 . . . . . 6  |-  F/_ x A
18 dff13f.1 . . . . . . . . 9  |-  F/_ x F
19 nfcv 2421 . . . . . . . . 9  |-  F/_ x w
2018, 19nffv 5534 . . . . . . . 8  |-  F/_ x
( F `  w
)
21 nfcv 2421 . . . . . . . . 9  |-  F/_ x
y
2218, 21nffv 5534 . . . . . . . 8  |-  F/_ x
( F `  y
)
2320, 22nfeq 2428 . . . . . . 7  |-  F/ x
( F `  w
)  =  ( F `
 y )
24 nfv 1607 . . . . . . 7  |-  F/ x  w  =  y
2523, 24nfim 1771 . . . . . 6  |-  F/ x
( ( F `  w )  =  ( F `  y )  ->  w  =  y )
2617, 25nfral 2598 . . . . 5  |-  F/ x A. y  e.  A  ( ( F `  w )  =  ( F `  y )  ->  w  =  y )
27 nfv 1607 . . . . 5  |-  F/ w A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y )
28 fveq2 5527 . . . . . . . 8  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
2928eqeq1d 2293 . . . . . . 7  |-  ( w  =  x  ->  (
( F `  w
)  =  ( F `
 y )  <->  ( F `  x )  =  ( F `  y ) ) )
30 eqeq1 2291 . . . . . . 7  |-  ( w  =  x  ->  (
w  =  y  <->  x  =  y ) )
3129, 30imbi12d 311 . . . . . 6  |-  ( w  =  x  ->  (
( ( F `  w )  =  ( F `  y )  ->  w  =  y )  <->  ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) ) )
3231ralbidv 2565 . . . . 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 2762 . . . 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 240 . . 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 675 . 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 240 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    <-> wb 176    /\ wa 358    = wceq 1625   F/_wnfc 2408   A.wral 2545   -->wf 5253   -1-1->wf1 5254   ` cfv 5257
This theorem is referenced by:  f1mpt  5787  dom2lem  6903  dff1o6f  25103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fv 5265
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