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Theorem dff13f 5636
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
StepHypRef Expression
1 dff13 5635 . 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 2385 . . . . . . . . 9  |-  F/_ y
w
42, 3nffv 5384 . . . . . . . 8  |-  F/_ y
( F `  w
)
5 nfcv 2385 . . . . . . . . 9  |-  F/_ y
v
62, 5nffv 5384 . . . . . . . 8  |-  F/_ y
( F `  v
)
74, 6nfeq 2392 . . . . . . 7  |-  F/ y ( F `  w
)  =  ( F `
 v )
8 nfv 1629 . . . . . . 7  |-  F/ y  w  =  v
97, 8nfim 1735 . . . . . 6  |-  F/ y ( ( F `  w )  =  ( F `  v )  ->  w  =  v )
10 nfv 1629 . . . . . 6  |-  F/ v ( ( F `  w )  =  ( F `  y )  ->  w  =  y )
11 fveq2 5377 . . . . . . . 8  |-  ( v  =  y  ->  ( F `  v )  =  ( F `  y ) )
1211eqeq2d 2264 . . . . . . 7  |-  ( v  =  y  ->  (
( F `  w
)  =  ( F `
 v )  <->  ( F `  w )  =  ( F `  y ) ) )
13 eqeq2 2262 . . . . . . 7  |-  ( v  =  y  ->  (
w  =  v  <->  w  =  y ) )
1412, 13imbi12d 313 . . . . . 6  |-  ( v  =  y  ->  (
( ( F `  w )  =  ( F `  v )  ->  w  =  v )  <->  ( ( F `
 w )  =  ( F `  y
)  ->  w  =  y ) ) )
159, 10, 14cbvral 2705 . . . . 5  |-  ( A. v  e.  A  (
( F `  w
)  =  ( F `
 v )  ->  w  =  v )  <->  A. y  e.  A  ( ( F `  w
)  =  ( F `
 y )  ->  w  =  y )
)
1615ralbii 2531 . . . 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 2385 . . . . . 6  |-  F/_ x A
18 dff13f.1 . . . . . . . . 9  |-  F/_ x F
19 nfcv 2385 . . . . . . . . 9  |-  F/_ x w
2018, 19nffv 5384 . . . . . . . 8  |-  F/_ x
( F `  w
)
21 nfcv 2385 . . . . . . . . 9  |-  F/_ x
y
2218, 21nffv 5384 . . . . . . . 8  |-  F/_ x
( F `  y
)
2320, 22nfeq 2392 . . . . . . 7  |-  F/ x
( F `  w
)  =  ( F `
 y )
24 nfv 1629 . . . . . . 7  |-  F/ x  w  =  y
2523, 24nfim 1735 . . . . . 6  |-  F/ x
( ( F `  w )  =  ( F `  y )  ->  w  =  y )
2617, 25nfral 2558 . . . . 5  |-  F/ x A. y  e.  A  ( ( F `  w )  =  ( F `  y )  ->  w  =  y )
27 nfv 1629 . . . . 5  |-  F/ w A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y )
28 fveq2 5377 . . . . . . . 8  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
2928eqeq1d 2261 . . . . . . 7  |-  ( w  =  x  ->  (
( F `  w
)  =  ( F `
 y )  <->  ( F `  x )  =  ( F `  y ) ) )
30 eqeq1 2259 . . . . . . 7  |-  ( w  =  x  ->  (
w  =  y  <->  x  =  y ) )
3129, 30imbi12d 313 . . . . . 6  |-  ( w  =  x  ->  (
( ( F `  w )  =  ( F `  y )  ->  w  =  y )  <->  ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) ) )
3231ralbidv 2527 . . . . 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 2705 . . . 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 242 . . 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 678 . 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 242 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 6    <-> wb 178    /\ wa 360    = wceq 1619   F/_wnfc 2372   A.wral 2509   -->wf 4588   -1-1->wf1 4589   ` cfv 4592
This theorem is referenced by:  f1mpt  5637  dom2lem  6787  dff1o6f  24257
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fv 4608
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