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Theorem eufnfv 5714
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 5708 . . . 4  |-  ( x  e.  A  |->  B )  e.  _V
3 eqeq2 2293 . . . . . 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 2876 . . 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 2284 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
97, 8fnmpti 5338 . . . . . 6  |-  ( x  e.  A  |->  B )  Fn  A
10 fneq1 5299 . . . . . 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 5530 . . . . . . 7  |-  ( f  Fn  A  <->  f  =  ( x  e.  A  |->  ( f `  x
) ) )
14 eqeq1 2290 . . . . . . 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 5500 . . . . . . . 8  |-  ( f `
 x )  e. 
_V
1716rgenw 2611 . . . . . . 7  |-  A. x  e.  A  ( f `  x )  e.  _V
18 mpteqb 5576 . . . . . . 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 2148 . 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 1685   E!weu 2144   A.wral 2544   _Vcvv 2789    e. cmpt 4078    Fn wfn 5216   ` cfv 5221
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229
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