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Theorem fvdiagfn 7017
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 448 . 2  |-  ( ( I  e.  W  /\  X  e.  B )  ->  X  e.  B )
2 snex 4365 . . . 4  |-  { X }  e.  _V
3 xpexg 4948 . . . 4  |-  ( ( I  e.  W  /\  { X }  e.  _V )  ->  ( I  X.  { X } )  e. 
_V )
42, 3mpan2 653 . . 3  |-  ( I  e.  W  ->  (
I  X.  { X } )  e.  _V )
54adantr 452 . 2  |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( I  X.  { X } )  e.  _V )
6 sneq 3785 . . . 4  |-  ( x  =  X  ->  { x }  =  { X } )
76xpeq2d 4861 . . 3  |-  ( x  =  X  ->  (
I  X.  { x } )  =  ( I  X.  { X } ) )
8 fdiagfn.f . . 3  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
97, 8fvmptg 5763 . 2  |-  ( ( X  e.  B  /\  ( I  X.  { X } )  e.  _V )  ->  ( F `  X )  =  ( I  X.  { X } ) )
101, 5, 9syl2anc 643 1  |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( F `  X
)  =  ( I  X.  { X }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   {csn 3774    e. cmpt 4226    X. cxp 4835   ` cfv 5413
This theorem is referenced by:  pwsdiagmhm  14723  pwsdiaglmhm  16088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421
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