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Theorem eufnfv 6125
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 6122 . . . 4  |-  ( x  e.  A  |->  B )  e.  _V
3 eqeq2 2475 . . . . . 6  |-  ( z  =  ( x  e.  A  |->  B )  -> 
( f  =  z  <-> 
f  =  ( x  e.  A  |->  B ) ) )
43bibi2d 318 . . . . 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 1684 . . . 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 3198 . . 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 2460 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
97, 8fnmpti 5700 . . . . . 6  |-  ( x  e.  A  |->  B )  Fn  A
10 fneq1 5660 . . . . . 6  |-  ( f  =  ( x  e.  A  |->  B )  -> 
( f  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
119, 10mpbiri 233 . . . . 5  |-  ( f  =  ( x  e.  A  |->  B )  -> 
f  Fn  A )
1211pm4.71ri 633 . . . 4  |-  ( f  =  ( x  e.  A  |->  B )  <->  ( f  Fn  A  /\  f  =  ( x  e.  A  |->  B ) ) )
13 dffn5 5904 . . . . . . 7  |-  ( f  Fn  A  <->  f  =  ( x  e.  A  |->  ( f `  x
) ) )
14 eqeq1 2464 . . . . . . 7  |-  ( f  =  ( x  e.  A  |->  ( f `  x ) )  -> 
( f  =  ( x  e.  A  |->  B )  <->  ( x  e.  A  |->  ( f `  x ) )  =  ( x  e.  A  |->  B ) ) )
1513, 14sylbi 195 . . . . . 6  |-  ( f  Fn  A  ->  (
f  =  ( x  e.  A  |->  B )  <-> 
( x  e.  A  |->  ( f `  x
) )  =  ( x  e.  A  |->  B ) ) )
16 fvex 5867 . . . . . . . 8  |-  ( f `
 x )  e. 
_V
1716rgenw 2818 . . . . . . 7  |-  A. x  e.  A  ( f `  x )  e.  _V
18 mpteqb 5955 . . . . . . 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 5 . . . . . 6  |-  ( ( x  e.  A  |->  ( f `  x ) )  =  ( x  e.  A  |->  B )  <->  A. x  e.  A  ( f `  x
)  =  B )
2015, 19syl6bb 261 . . . . 5  |-  ( f  Fn  A  ->  (
f  =  ( x  e.  A  |->  B )  <->  A. x  e.  A  ( f `  x
)  =  B ) )
2120pm5.32i 637 . . . 4  |-  ( ( f  Fn  A  /\  f  =  ( x  e.  A  |->  B ) )  <->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B ) )
2212, 21bitr2i 250 . . 3  |-  ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B )  <->  f  =  ( x  e.  A  |->  B ) )
236, 22mpg 1598 . 2  |-  E. z A. f ( ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  =  B )  <->  f  =  z )
24 df-eu 2272 . 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 209 1  |-  E! f ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   A.wal 1372    = wceq 1374   E.wex 1591    e. wcel 1762   E!weu 2268   A.wral 2807   _Vcvv 3106    |-> cmpt 4498    Fn wfn 5574   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587
This theorem is referenced by: (None)
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