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Theorem dffun4 5466
Description: Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
dffun4  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x A. y A. z ( ( <.
x ,  y >.  e.  A  /\  <. x ,  z >.  e.  A
)  ->  y  =  z ) ) )
Distinct variable group:    x, y, z, A

Proof of Theorem dffun4
StepHypRef Expression
1 dffun2 5464 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x A. y A. z ( ( x A y  /\  x A z )  -> 
y  =  z ) ) )
2 df-br 4213 . . . . . . 7  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
3 df-br 4213 . . . . . . 7  |-  ( x A z  <->  <. x ,  z >.  e.  A
)
42, 3anbi12i 679 . . . . . 6  |-  ( ( x A y  /\  x A z )  <->  ( <. x ,  y >.  e.  A  /\  <. x ,  z
>.  e.  A ) )
54imbi1i 316 . . . . 5  |-  ( ( ( x A y  /\  x A z )  ->  y  =  z )  <->  ( ( <. x ,  y >.  e.  A  /\  <. x ,  z >.  e.  A
)  ->  y  =  z ) )
65albii 1575 . . . 4  |-  ( A. z ( ( x A y  /\  x A z )  -> 
y  =  z )  <->  A. z ( ( <.
x ,  y >.  e.  A  /\  <. x ,  z >.  e.  A
)  ->  y  =  z ) )
762albii 1576 . . 3  |-  ( A. x A. y A. z
( ( x A y  /\  x A z )  ->  y  =  z )  <->  A. x A. y A. z ( ( <. x ,  y
>.  e.  A  /\  <. x ,  z >.  e.  A
)  ->  y  =  z ) )
87anbi2i 676 . 2  |-  ( ( Rel  A  /\  A. x A. y A. z
( ( x A y  /\  x A z )  ->  y  =  z ) )  <-> 
( Rel  A  /\  A. x A. y A. z ( ( <.
x ,  y >.  e.  A  /\  <. x ,  z >.  e.  A
)  ->  y  =  z ) ) )
91, 8bitri 241 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x A. y A. z ( ( <.
x ,  y >.  e.  A  /\  <. x ,  z >.  e.  A
)  ->  y  =  z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    e. wcel 1725   <.cop 3817   class class class wbr 4212   Rel wrel 4883   Fun wfun 5448
This theorem is referenced by:  funopg  5485  funun  5495  fununi  5517  tfrlem7  6644  hashfun  11700  dffun10  25759  elfuns  25760  bnj1379  29202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-cnv 4886  df-co 4887  df-fun 5456
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