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Theorem eufnfv 5912
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 5906 . . . 4  |-  ( x  e.  A  |->  B )  e.  _V
3 eqeq2 2397 . . . . . 6  |-  ( z  =  ( x  e.  A  |->  B )  -> 
( f  =  z  <-> 
f  =  ( x  e.  A  |->  B ) ) )
43bibi2d 310 . . . . 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 1632 . . . 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 2987 . . 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 2388 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
97, 8fnmpti 5514 . . . . . 6  |-  ( x  e.  A  |->  B )  Fn  A
10 fneq1 5475 . . . . . 6  |-  ( f  =  ( x  e.  A  |->  B )  -> 
( f  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
119, 10mpbiri 225 . . . . 5  |-  ( f  =  ( x  e.  A  |->  B )  -> 
f  Fn  A )
1211pm4.71ri 615 . . . 4  |-  ( f  =  ( x  e.  A  |->  B )  <->  ( f  Fn  A  /\  f  =  ( x  e.  A  |->  B ) ) )
13 dffn5 5712 . . . . . . 7  |-  ( f  Fn  A  <->  f  =  ( x  e.  A  |->  ( f `  x
) ) )
14 eqeq1 2394 . . . . . . 7  |-  ( f  =  ( x  e.  A  |->  ( f `  x ) )  -> 
( f  =  ( x  e.  A  |->  B )  <->  ( x  e.  A  |->  ( f `  x ) )  =  ( x  e.  A  |->  B ) ) )
1513, 14sylbi 188 . . . . . 6  |-  ( f  Fn  A  ->  (
f  =  ( x  e.  A  |->  B )  <-> 
( x  e.  A  |->  ( f `  x
) )  =  ( x  e.  A  |->  B ) ) )
16 fvex 5683 . . . . . . . 8  |-  ( f `
 x )  e. 
_V
1716rgenw 2717 . . . . . . 7  |-  A. x  e.  A  ( f `  x )  e.  _V
18 mpteqb 5759 . . . . . . 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 253 . . . . 5  |-  ( f  Fn  A  ->  (
f  =  ( x  e.  A  |->  B )  <->  A. x  e.  A  ( f `  x
)  =  B ) )
2120pm5.32i 619 . . . 4  |-  ( ( f  Fn  A  /\  f  =  ( x  e.  A  |->  B ) )  <->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B ) )
2212, 21bitr2i 242 . . 3  |-  ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  =  B )  <->  f  =  ( x  e.  A  |->  B ) )
236, 22mpg 1554 . 2  |-  E. z A. f ( ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  =  B )  <->  f  =  z )
24 df-eu 2243 . 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 201 1  |-  E! f ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   E!weu 2239   A.wral 2650   _Vcvv 2900    e. cmpt 4208    Fn wfn 5390   ` cfv 5395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403
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