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Theorem fvdiagfn 7343
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 461 . 2  |-  ( ( I  e.  W  /\  X  e.  B )  ->  X  e.  B )
2 snex 4617 . . . 4  |-  { X }  e.  _V
3 xpexg 6593 . . . 4  |-  ( ( I  e.  W  /\  { X }  e.  _V )  ->  ( I  X.  { X } )  e. 
_V )
42, 3mpan2 671 . . 3  |-  ( I  e.  W  ->  (
I  X.  { X } )  e.  _V )
54adantr 465 . 2  |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( I  X.  { X } )  e.  _V )
6 sneq 3971 . . . 4  |-  ( x  =  X  ->  { x }  =  { X } )
76xpeq2d 4948 . . 3  |-  ( x  =  X  ->  (
I  X.  { x } )  =  ( I  X.  { X } ) )
8 fdiagfn.f . . 3  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
97, 8fvmptg 5857 . 2  |-  ( ( X  e.  B  /\  ( I  X.  { X } )  e.  _V )  ->  ( F `  X )  =  ( I  X.  { X } ) )
101, 5, 9syl2anc 661 1  |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( F `  X
)  =  ( I  X.  { X }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757   _Vcvv 3054   {csn 3961    |-> cmpt 4434    X. cxp 4922   ` cfv 5502
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-iota 5465  df-fun 5504  df-fv 5510
This theorem is referenced by:  pwsdiagmhm  15585  pwsdiaglmhm  17230
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