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Theorem eufnfv 5752
Description: A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
Hypotheses
Ref Expression
eufnfv.1  |-  A  e. 
_V
eufnfv.2  |-  B  e. 
_V
Assertion
Ref Expression
eufnfv  |-  E! f ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  =  B )
Distinct variable groups:    x, f, A    B, f
Allowed substitution hint:    B( x)

Proof of Theorem eufnfv
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eufnfv.1 . . . . 5  |-  A  e. 
_V
21mptex 5746 . . . 4  |-  ( x  e.  A  |->  B )  e.  _V
3 eqeq2 2292 . . . . . 6  |-  ( z  =  ( x  e.  A  |->  B )  -> 
( f  =  z  <-> 
f  =  ( x  e.  A  |->  B ) ) )
43bibi2d 309 . . . . 5  |-  ( z  =  ( x  e.  A  |->  B )  -> 
( ( ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  =  B )  <->  f  =  z )  <->  ( (
f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B )  <->  f  =  ( x  e.  A  |->  B ) ) ) )
54albidv 1611 . . . 4  |-  ( z  =  ( x  e.  A  |->  B )  -> 
( A. f ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  =  B )  <-> 
f  =  z )  <->  A. f ( ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  =  B )  <->  f  =  ( x  e.  A  |->  B ) ) ) )
62, 5spcev 2875 . . 3  |-  ( A. f ( ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  =  B )  <->  f  =  ( x  e.  A  |->  B ) )  ->  E. z A. f ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  =  B )  <-> 
f  =  z ) )
7 eufnfv.2 . . . . . . 7  |-  B  e. 
_V
8 eqid 2283 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
97, 8fnmpti 5372 . . . . . 6  |-  ( x  e.  A  |->  B )  Fn  A
10 fneq1 5333 . . . . . 6  |-  ( f  =  ( x  e.  A  |->  B )  -> 
( f  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
119, 10mpbiri 224 . . . . 5  |-  ( f  =  ( x  e.  A  |->  B )  -> 
f  Fn  A )
1211pm4.71ri 614 . . . 4  |-  ( f  =  ( x  e.  A  |->  B )  <->  ( f  Fn  A  /\  f  =  ( x  e.  A  |->  B ) ) )
13 dffn5 5568 . . . . . . 7  |-  ( f  Fn  A  <->  f  =  ( x  e.  A  |->  ( f `  x
) ) )
14 eqeq1 2289 . . . . . . 7  |-  ( f  =  ( x  e.  A  |->  ( f `  x ) )  -> 
( f  =  ( x  e.  A  |->  B )  <->  ( x  e.  A  |->  ( f `  x ) )  =  ( x  e.  A  |->  B ) ) )
1513, 14sylbi 187 . . . . . 6  |-  ( f  Fn  A  ->  (
f  =  ( x  e.  A  |->  B )  <-> 
( x  e.  A  |->  ( f `  x
) )  =  ( x  e.  A  |->  B ) ) )
16 fvex 5539 . . . . . . . 8  |-  ( f `
 x )  e. 
_V
1716rgenw 2610 . . . . . . 7  |-  A. x  e.  A  ( f `  x )  e.  _V
18 mpteqb 5614 . . . . . . 7  |-  ( A. x  e.  A  (
f `  x )  e.  _V  ->  ( (
x  e.  A  |->  ( f `  x ) )  =  ( x  e.  A  |->  B )  <->  A. x  e.  A  ( f `  x
)  =  B ) )
1917, 18ax-mp 8 . . . . . 6  |-  ( ( x  e.  A  |->  ( f `  x ) )  =  ( x  e.  A  |->  B )  <->  A. x  e.  A  ( f `  x
)  =  B )
2015, 19syl6bb 252 . . . . 5  |-  ( f  Fn  A  ->  (
f  =  ( x  e.  A  |->  B )  <->  A. x  e.  A  ( f `  x
)  =  B ) )
2120pm5.32i 618 . . . 4  |-  ( ( f  Fn  A  /\  f  =  ( x  e.  A  |->  B ) )  <->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B ) )
2212, 21bitr2i 241 . . 3  |-  ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B )  <->  f  =  ( x  e.  A  |->  B ) )
236, 22mpg 1535 . 2  |-  E. z A. f ( ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  =  B )  <->  f  =  z )
24 df-eu 2147 . 2  |-  ( E! f ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B )  <->  E. z A. f
( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B )  <->  f  =  z ) )
2523, 24mpbir 200 1  |-  E! f ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   A.wral 2543   _Vcvv 2788    e. cmpt 4077    Fn wfn 5250   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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