MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fdiagfn Unicode version

Theorem fdiagfn 6811
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 5430 . . . 4  |-  ( x  e.  B  ->  (
I  X.  { x } ) : I --> B )
21adantl 452 . . 3  |-  ( ( ( B  e.  V  /\  I  e.  W
)  /\  x  e.  B )  ->  (
I  X.  { x } ) : I --> B )
3 elmapg 6785 . . . 4  |-  ( ( B  e.  V  /\  I  e.  W )  ->  ( ( I  X.  { x } )  e.  ( B  ^m  I )  <->  ( I  X.  { x } ) : I --> B ) )
43adantr 451 . . 3  |-  ( ( ( B  e.  V  /\  I  e.  W
)  /\  x  e.  B )  ->  (
( I  X.  {
x } )  e.  ( B  ^m  I
)  <->  ( I  X.  { x } ) : I --> B ) )
52, 4mpbird 223 . 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 5684 1  |-  ( ( B  e.  V  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640    e. cmpt 4077    X. cxp 4687   -->wf 5251  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  pwsdiagmhm  14445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774
  Copyright terms: Public domain W3C validator