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Theorem fvdiagfn 7456
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 459 . 2  |-  ( ( I  e.  W  /\  X  e.  B )  ->  X  e.  B )
2 snex 4678 . . . 4  |-  { X }  e.  _V
3 xpexg 6575 . . . 4  |-  ( ( I  e.  W  /\  { X }  e.  _V )  ->  ( I  X.  { X } )  e. 
_V )
42, 3mpan2 669 . . 3  |-  ( I  e.  W  ->  (
I  X.  { X } )  e.  _V )
54adantr 463 . 2  |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( I  X.  { X } )  e.  _V )
6 sneq 4026 . . . 4  |-  ( x  =  X  ->  { x }  =  { X } )
76xpeq2d 5012 . . 3  |-  ( x  =  X  ->  (
I  X.  { x } )  =  ( I  X.  { X } ) )
8 fdiagfn.f . . 3  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
97, 8fvmptg 5929 . 2  |-  ( ( X  e.  B  /\  ( I  X.  { X } )  e.  _V )  ->  ( F `  X )  =  ( I  X.  { X } ) )
101, 5, 9syl2anc 659 1  |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( F `  X
)  =  ( I  X.  { X }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016    |-> cmpt 4497    X. cxp 4986   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578
This theorem is referenced by:  pwsdiagmhm  16202  pwsdiaglmhm  17901
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