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Theorem fvdiagfn 7087
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
fvdiagfn  |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( F `  X
)  =  ( I  X.  { X }
) )
Distinct variable groups:    x, B    x, I    x, W    x, X
Allowed substitution hint:    F( x)

Proof of Theorem fvdiagfn
StepHypRef Expression
1 simpr 449 . 2  |-  ( ( I  e.  W  /\  X  e.  B )  ->  X  e.  B )
2 snex 4434 . . . 4  |-  { X }  e.  _V
3 xpexg 5018 . . . 4  |-  ( ( I  e.  W  /\  { X }  e.  _V )  ->  ( I  X.  { X } )  e. 
_V )
42, 3mpan2 654 . . 3  |-  ( I  e.  W  ->  (
I  X.  { X } )  e.  _V )
54adantr 453 . 2  |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( I  X.  { X } )  e.  _V )
6 sneq 3849 . . . 4  |-  ( x  =  X  ->  { x }  =  { X } )
76xpeq2d 4931 . . 3  |-  ( x  =  X  ->  (
I  X.  { x } )  =  ( I  X.  { X } ) )
8 fdiagfn.f . . 3  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
97, 8fvmptg 5833 . 2  |-  ( ( X  e.  B  /\  ( I  X.  { X } )  e.  _V )  ->  ( F `  X )  =  ( I  X.  { X } ) )
101, 5, 9syl2anc 644 1  |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( F `  X
)  =  ( I  X.  { X }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   _Vcvv 2962   {csn 3838    e. cmpt 4291    X. cxp 4905   ` cfv 5483
This theorem is referenced by:  pwsdiagmhm  14799  pwsdiaglmhm  16164
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-iota 5447  df-fun 5485  df-fv 5491
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