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Theorem fvdiagfn 7460
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 4688 . . . 4  |-  { X }  e.  _V
3 xpexg 6709 . . . 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 4037 . . . 4  |-  ( x  =  X  ->  { x }  =  { X } )
76xpeq2d 5023 . . 3  |-  ( x  =  X  ->  (
I  X.  { x } )  =  ( I  X.  { X } ) )
8 fdiagfn.f . . 3  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
97, 8fvmptg 5946 . 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 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    |-> cmpt 4505    X. cxp 4997   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594
This theorem is referenced by:  pwsdiagmhm  15807  pwsdiaglmhm  17483
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