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Theorem mpt2sn 6887
Description: An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)
Hypotheses
Ref Expression
mpt2sn.f  |-  F  =  ( x  e.  { A } ,  y  e. 
{ B }  |->  C )
mpt2sn.a  |-  ( x  =  A  ->  C  =  D )
mpt2sn.b  |-  ( y  =  B  ->  D  =  E )
Assertion
Ref Expression
mpt2sn  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  F  =  { <. <. A ,  B >. ,  E >. } )
Distinct variable groups:    x, A, y    x, B, y    x, E, y    x, U, y   
x, V, y    x, W, y
Allowed substitution hints:    C( x, y)    D( x, y)    F( x, y)

Proof of Theorem mpt2sn
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 xpsng 6065 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  { B } )  =  { <. A ,  B >. } )
213adant3 1028 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  ( { A }  X.  { B } )  =  { <. A ,  B >. } )
32mpteq1d 4484 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  ( p  e.  ( { A }  X.  { B } )  |->  [_ ( 1st `  p )  /  x ]_ [_ ( 2nd `  p )  / 
y ]_ C )  =  ( p  e.  { <. A ,  B >. } 
|->  [_ ( 1st `  p
)  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C ) )
4 mpt2sn.f . . . 4  |-  F  =  ( x  e.  { A } ,  y  e. 
{ B }  |->  C )
5 mpt2mpts 6857 . . . 4  |-  ( x  e.  { A } ,  y  e.  { B }  |->  C )  =  ( p  e.  ( { A }  X.  { B } )  |->  [_ ( 1st `  p )  /  x ]_ [_ ( 2nd `  p )  / 
y ]_ C )
64, 5eqtri 2473 . . 3  |-  F  =  ( p  e.  ( { A }  X.  { B } )  |->  [_ ( 1st `  p )  /  x ]_ [_ ( 2nd `  p )  / 
y ]_ C )
76a1i 11 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  F  =  ( p  e.  ( { A }  X.  { B }
)  |->  [_ ( 1st `  p
)  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C ) )
8 op2ndg 6806 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
9 fveq2 5865 . . . . . . . . 9  |-  ( p  =  <. A ,  B >.  ->  ( 2nd `  p
)  =  ( 2nd `  <. A ,  B >. ) )
109eqcomd 2457 . . . . . . . 8  |-  ( p  =  <. A ,  B >.  ->  ( 2nd `  <. A ,  B >. )  =  ( 2nd `  p
) )
1110eqeq1d 2453 . . . . . . 7  |-  ( p  =  <. A ,  B >.  ->  ( ( 2nd `  <. A ,  B >. )  =  B  <->  ( 2nd `  p )  =  B ) )
128, 11syl5ibcom 224 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( p  =  <. A ,  B >.  ->  ( 2nd `  p )  =  B ) )
13123adant3 1028 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  ( p  =  <. A ,  B >.  ->  ( 2nd `  p )  =  B ) )
1413imp 431 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  p  =  <. A ,  B >. )  ->  ( 2nd `  p
)  =  B )
15 op1stg 6805 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
16 fveq2 5865 . . . . . . . . 9  |-  ( p  =  <. A ,  B >.  ->  ( 1st `  p
)  =  ( 1st `  <. A ,  B >. ) )
1716eqcomd 2457 . . . . . . . 8  |-  ( p  =  <. A ,  B >.  ->  ( 1st `  <. A ,  B >. )  =  ( 1st `  p
) )
1817eqeq1d 2453 . . . . . . 7  |-  ( p  =  <. A ,  B >.  ->  ( ( 1st `  <. A ,  B >. )  =  A  <->  ( 1st `  p )  =  A ) )
1915, 18syl5ibcom 224 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( p  =  <. A ,  B >.  ->  ( 1st `  p )  =  A ) )
20193adant3 1028 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  ( p  =  <. A ,  B >.  ->  ( 1st `  p )  =  A ) )
2120imp 431 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  p  =  <. A ,  B >. )  ->  ( 1st `  p
)  =  A )
22 simp1 1008 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  A  e.  V )
23 simpl2 1012 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  x  =  A )  ->  B  e.  W )
24 mpt2sn.a . . . . . . . . . 10  |-  ( x  =  A  ->  C  =  D )
2524adantl 468 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  x  =  A )  ->  C  =  D )
26 mpt2sn.b . . . . . . . . 9  |-  ( y  =  B  ->  D  =  E )
2725, 26sylan9eq 2505 . . . . . . . 8  |-  ( ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  /\  x  =  A )  /\  y  =  B )  ->  C  =  E )
2823, 27csbied 3390 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  x  =  A )  ->  [_ B  /  y ]_ C  =  E )
2922, 28csbied 3390 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  E
)
3029adantr 467 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  p  =  <. A ,  B >. )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  E
)
31 csbeq1 3366 . . . . . . . 8  |-  ( ( 1st `  p )  =  A  ->  [_ ( 1st `  p )  /  x ]_ [_ ( 2nd `  p )  /  y ]_ C  =  [_ A  /  x ]_ [_ ( 2nd `  p )  / 
y ]_ C )
3231eqeq1d 2453 . . . . . . 7  |-  ( ( 1st `  p )  =  A  ->  ( [_ ( 1st `  p
)  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  E  <->  [_ A  /  x ]_ [_ ( 2nd `  p )  /  y ]_ C  =  E
) )
3332adantl 468 . . . . . 6  |-  ( ( ( 2nd `  p
)  =  B  /\  ( 1st `  p )  =  A )  -> 
( [_ ( 1st `  p
)  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  E  <->  [_ A  /  x ]_ [_ ( 2nd `  p )  /  y ]_ C  =  E
) )
34 csbeq1 3366 . . . . . . . . 9  |-  ( ( 2nd `  p )  =  B  ->  [_ ( 2nd `  p )  / 
y ]_ C  =  [_ B  /  y ]_ C
)
3534adantr 467 . . . . . . . 8  |-  ( ( ( 2nd `  p
)  =  B  /\  ( 1st `  p )  =  A )  ->  [_ ( 2nd `  p
)  /  y ]_ C  =  [_ B  / 
y ]_ C )
3635csbeq2dv 3781 . . . . . . 7  |-  ( ( ( 2nd `  p
)  =  B  /\  ( 1st `  p )  =  A )  ->  [_ A  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  [_ A  /  x ]_ [_ B  / 
y ]_ C )
3736eqeq1d 2453 . . . . . 6  |-  ( ( ( 2nd `  p
)  =  B  /\  ( 1st `  p )  =  A )  -> 
( [_ A  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  E  <->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  E ) )
3833, 37bitrd 257 . . . . 5  |-  ( ( ( 2nd `  p
)  =  B  /\  ( 1st `  p )  =  A )  -> 
( [_ ( 1st `  p
)  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  E  <->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  E ) )
3930, 38syl5ibrcom 226 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  p  =  <. A ,  B >. )  ->  ( ( ( 2nd `  p )  =  B  /\  ( 1st `  p )  =  A )  ->  [_ ( 1st `  p )  /  x ]_ [_ ( 2nd `  p )  /  y ]_ C  =  E
) )
4014, 21, 39mp2and 685 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  p  =  <. A ,  B >. )  ->  [_ ( 1st `  p
)  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  E )
41 opex 4664 . . . 4  |-  <. A ,  B >.  e.  _V
4241a1i 11 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  -> 
<. A ,  B >.  e. 
_V )
43 simp3 1010 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  E  e.  U )
4440, 42, 43fmptsnd 6086 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  { <. <. A ,  B >. ,  E >. }  =  ( p  e.  { <. A ,  B >. }  |->  [_ ( 1st `  p )  /  x ]_ [_ ( 2nd `  p )  / 
y ]_ C ) )
453, 7, 443eqtr4d 2495 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  F  =  { <. <. A ,  B >. ,  E >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   _Vcvv 3045   [_csb 3363   {csn 3968   <.cop 3974    |-> cmpt 4461    X. cxp 4832   ` cfv 5582    |-> cmpt2 6292   1stc1st 6791   2ndc2nd 6792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794
This theorem is referenced by:  mat1dim0  19498  mat1dimid  19499  mat1dimmul  19501  d1mat2pmat  19763
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