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Theorem bj-diagval 33894
Description: Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-diagval  |-  ( A  e.  V  ->  (Diag `  A )  =  (  _I  i^i  ( A  X.  A ) ) )

Proof of Theorem bj-diagval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 3122 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 incom 3691 . . 3  |-  ( ( A  X.  A )  i^i  _I  )  =  (  _I  i^i  ( A  X.  A ) )
3 xpexg 6587 . . . . 5  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
41, 1, 3syl2anc 661 . . . 4  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
5 inex1g 4590 . . . 4  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  i^i  _I  )  e.  _V )
64, 5syl 16 . . 3  |-  ( A  e.  V  ->  (
( A  X.  A
)  i^i  _I  )  e.  _V )
72, 6syl5eqelr 2560 . 2  |-  ( A  e.  V  ->  (  _I  i^i  ( A  X.  A ) )  e. 
_V )
8 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
98, 8xpeq12d 5024 . . . 4  |-  ( x  =  A  ->  (
x  X.  x )  =  ( A  X.  A ) )
109ineq2d 3700 . . 3  |-  ( x  =  A  ->  (  _I  i^i  ( x  X.  x ) )  =  (  _I  i^i  ( A  X.  A ) ) )
11 df-bj-diag 33893 . . 3  |- Diag  =  ( x  e.  _V  |->  (  _I  i^i  ( x  X.  x ) ) )
1210, 11fvmptg 5949 . 2  |-  ( ( A  e.  _V  /\  (  _I  i^i  ( A  X.  A ) )  e.  _V )  -> 
(Diag `  A )  =  (  _I  i^i  ( A  X.  A
) ) )
131, 7, 12syl2anc 661 1  |-  ( A  e.  V  ->  (Diag `  A )  =  (  _I  i^i  ( A  X.  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    _I cid 4790    X. cxp 4997   ` cfv 5588  Diagcdiag2 33892
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 6577
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 5551  df-fun 5590  df-fv 5596  df-bj-diag 33893
This theorem is referenced by:  bj-eldiag  33895
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