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Theorem fdiagfn 7086
Description: Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypothesis
Ref Expression
fdiagfn.f  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
Assertion
Ref Expression
fdiagfn  |-  ( ( B  e.  V  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
Distinct variable groups:    x, B    x, I    x, V    x, W
Allowed substitution hint:    F( x)

Proof of Theorem fdiagfn
StepHypRef Expression
1 fconst6g 5661 . . . 4  |-  ( x  e.  B  ->  (
I  X.  { x } ) : I --> B )
21adantl 454 . . 3  |-  ( ( ( B  e.  V  /\  I  e.  W
)  /\  x  e.  B )  ->  (
I  X.  { x } ) : I --> B )
3 elmapg 7060 . . . 4  |-  ( ( B  e.  V  /\  I  e.  W )  ->  ( ( I  X.  { x } )  e.  ( B  ^m  I )  <->  ( I  X.  { x } ) : I --> B ) )
43adantr 453 . . 3  |-  ( ( ( B  e.  V  /\  I  e.  W
)  /\  x  e.  B )  ->  (
( I  X.  {
x } )  e.  ( B  ^m  I
)  <->  ( I  X.  { x } ) : I --> B ) )
52, 4mpbird 225 . 2  |-  ( ( ( B  e.  V  /\  I  e.  W
)  /\  x  e.  B )  ->  (
I  X.  { x } )  e.  ( B  ^m  I ) )
6 fdiagfn.f . 2  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
75, 6fmptd 5922 1  |-  ( ( B  e.  V  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   {csn 3838    e. cmpt 4291    X. cxp 4905   -->wf 5479  (class class class)co 6110    ^m cmap 7047
This theorem is referenced by:  pwsdiagmhm  14799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-map 7049
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