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Theorem eufnfv 5507
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  y is distinct from all other variables.
StepHypRef Expression
1 eufnfv.1 . . . . 5  |-  A  e. 
_V
21mptex 5505 . . . 4  |-  ( x  e.  A  |->  B )  e.  _V
3 eqeq2 2097 . . . . . 6  |-  ( y  =  ( x  e.  A  |->  B )  -> 
( f  =  y  <-> 
f  =  ( x  e.  A  |->  B ) ) )
43bibi2d 230 . . . . 5  |-  ( y  =  ( x  e.  A  |->  B )  -> 
( ( ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  =  B )  <->  f  =  y )  <->  ( (
f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B )  <->  f  =  ( x  e.  A  |->  B ) ) ) )
54albidv 1752 . . . 4  |-  ( y  =  ( x  e.  A  |->  B )  -> 
( A. f ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  =  B )  <-> 
f  =  y )  <->  A. f ( ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  =  B )  <->  f  =  ( x  e.  A  |->  B ) ) ) )
62, 5spcev 2713 . . 3  |-  ( A. f ( ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  =  B )  <->  f  =  ( x  e.  A  |->  B ) )  ->  E. y A. f ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  =  B )  <-> 
f  =  y ) )
7 eufnfv.2 . . . . . . 7  |-  B  e. 
_V
8 eqid 2088 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
97, 8fnmpti 5128 . . . . . 6  |-  ( x  e.  A  |->  B )  Fn  A
10 fneq1 5088 . . . . . 6  |-  ( f  =  ( x  e.  A  |->  B )  -> 
( f  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
119, 10mpbiri 166 . . . . 5  |-  ( f  =  ( x  e.  A  |->  B )  -> 
f  Fn  A )
1211pm4.71ri 384 . . . 4  |-  ( f  =  ( x  e.  A  |->  B )  <->  ( f  Fn  A  /\  f  =  ( x  e.  A  |->  B ) ) )
13 dffn5im 5334 . . . . . . 7  |-  ( f  Fn  A  ->  f  =  ( x  e.  A  |->  ( f `  x ) ) )
1413eqeq1d 2096 . . . . . 6  |-  ( f  Fn  A  ->  (
f  =  ( x  e.  A  |->  B )  <-> 
( x  e.  A  |->  ( f `  x
) )  =  ( x  e.  A  |->  B ) ) )
15 funfvex 5306 . . . . . . . . 9  |-  ( ( Fun  f  /\  x  e.  dom  f )  -> 
( f `  x
)  e.  _V )
1615funfni 5100 . . . . . . . 8  |-  ( ( f  Fn  A  /\  x  e.  A )  ->  ( f `  x
)  e.  _V )
1716ralrimiva 2446 . . . . . . 7  |-  ( f  Fn  A  ->  A. x  e.  A  ( f `  x )  e.  _V )
18 mpteqb 5377 . . . . . . 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, 18syl 14 . . . . . 6  |-  ( f  Fn  A  ->  (
( x  e.  A  |->  ( f `  x
) )  =  ( x  e.  A  |->  B )  <->  A. x  e.  A  ( f `  x
)  =  B ) )
2014, 19bitrd 186 . . . . 5  |-  ( f  Fn  A  ->  (
f  =  ( x  e.  A  |->  B )  <->  A. x  e.  A  ( f `  x
)  =  B ) )
2120pm5.32i 442 . . . 4  |-  ( ( f  Fn  A  /\  f  =  ( x  e.  A  |->  B ) )  <->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B ) )
2212, 21bitr2i 183 . . 3  |-  ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B )  <->  f  =  ( x  e.  A  |->  B ) )
236, 22mpg 1385 . 2  |-  E. y A. f ( ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  =  B )  <->  f  =  y )
24 df-eu 1951 . 2  |-  ( E! f ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B )  <->  E. y A. f
( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B )  <->  f  =  y ) )
2523, 24mpbir 144 1  |-  E! f ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  =  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   E!weu 1948   A.wral 2359   _Vcvv 2619    |-> cmpt 3891    Fn wfn 4997   ` cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010
This theorem is referenced by: (None)
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