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Theorem dff13f 5906
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 5904 . 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 2502 . . . . . . . . 9  |-  F/_ y
w
42, 3nffv 5639 . . . . . . . 8  |-  F/_ y
( F `  w
)
5 nfcv 2502 . . . . . . . . 9  |-  F/_ y
v
62, 5nffv 5639 . . . . . . . 8  |-  F/_ y
( F `  v
)
74, 6nfeq 2509 . . . . . . 7  |-  F/ y ( F `  w
)  =  ( F `
 v )
8 nfv 1624 . . . . . . 7  |-  F/ y  w  =  v
97, 8nfim 1820 . . . . . 6  |-  F/ y ( ( F `  w )  =  ( F `  v )  ->  w  =  v )
10 nfv 1624 . . . . . 6  |-  F/ v ( ( F `  w )  =  ( F `  y )  ->  w  =  y )
11 fveq2 5632 . . . . . . . 8  |-  ( v  =  y  ->  ( F `  v )  =  ( F `  y ) )
1211eqeq2d 2377 . . . . . . 7  |-  ( v  =  y  ->  (
( F `  w
)  =  ( F `
 v )  <->  ( F `  w )  =  ( F `  y ) ) )
13 eqeq2 2375 . . . . . . 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 2845 . . . . 5  |-  ( A. v  e.  A  (
( F `  w
)  =  ( F `
 v )  ->  w  =  v )  <->  A. y  e.  A  ( ( F `  w
)  =  ( F `
 y )  ->  w  =  y )
)
1615ralbii 2652 . . . 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 2502 . . . . . 6  |-  F/_ x A
18 dff13f.1 . . . . . . . . 9  |-  F/_ x F
19 nfcv 2502 . . . . . . . . 9  |-  F/_ x w
2018, 19nffv 5639 . . . . . . . 8  |-  F/_ x
( F `  w
)
21 nfcv 2502 . . . . . . . . 9  |-  F/_ x
y
2218, 21nffv 5639 . . . . . . . 8  |-  F/_ x
( F `  y
)
2320, 22nfeq 2509 . . . . . . 7  |-  F/ x
( F `  w
)  =  ( F `
 y )
24 nfv 1624 . . . . . . 7  |-  F/ x  w  =  y
2523, 24nfim 1820 . . . . . 6  |-  F/ x
( ( F `  w )  =  ( F `  y )  ->  w  =  y )
2617, 25nfral 2681 . . . . 5  |-  F/ x A. y  e.  A  ( ( F `  w )  =  ( F `  y )  ->  w  =  y )
27 nfv 1624 . . . . 5  |-  F/ w A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y )
28 fveq2 5632 . . . . . . . 8  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
2928eqeq1d 2374 . . . . . . 7  |-  ( w  =  x  ->  (
( F `  w
)  =  ( F `
 y )  <->  ( F `  x )  =  ( F `  y ) ) )
30 eqeq1 2372 . . . . . . 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 2648 . . . . 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 2845 . . . 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 1647   F/_wnfc 2489   A.wral 2628   -->wf 5354   -1-1->wf1 5355   ` cfv 5358
This theorem is referenced by:  f1mpt  5907  dom2lem  7044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fv 5366
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