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Theorem fdiagfn 7359
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 5700 . . . 4  |-  ( x  e.  B  ->  (
I  X.  { x } ) : I --> B )
21adantl 466 . . 3  |-  ( ( ( B  e.  V  /\  I  e.  W
)  /\  x  e.  B )  ->  (
I  X.  { x } ) : I --> B )
3 elmapg 7330 . . . 4  |-  ( ( B  e.  V  /\  I  e.  W )  ->  ( ( I  X.  { x } )  e.  ( B  ^m  I )  <->  ( I  X.  { x } ) : I --> B ) )
43adantr 465 . . 3  |-  ( ( ( B  e.  V  /\  I  e.  W
)  /\  x  e.  B )  ->  (
( I  X.  {
x } )  e.  ( B  ^m  I
)  <->  ( I  X.  { x } ) : I --> B ) )
52, 4mpbird 232 . 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 5969 1  |-  ( ( B  e.  V  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {csn 3978    |-> cmpt 4451    X. cxp 4939   -->wf 5515  (class class class)co 6193    ^m cmap 7317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319
This theorem is referenced by:  pwsdiagmhm  15608
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