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Theorem mpt2sn 6890
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 6073 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  { B } )  =  { <. A ,  B >. } )
213adant3 1016 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  ( { A }  X.  { B } )  =  { <. A ,  B >. } )
32mpteq1d 4538 . 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 6863 . . . 4  |-  ( x  e.  { A } ,  y  e.  { B }  |->  C )  =  ( p  e.  ( { A }  X.  { B } )  |->  [_ ( 1st `  p )  /  x ]_ [_ ( 2nd `  p )  / 
y ]_ C )
64, 5eqtri 2486 . . 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 6812 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
9 fveq2 5872 . . . . . . . . 9  |-  ( p  =  <. A ,  B >.  ->  ( 2nd `  p
)  =  ( 2nd `  <. A ,  B >. ) )
109eqcomd 2465 . . . . . . . 8  |-  ( p  =  <. A ,  B >.  ->  ( 2nd `  <. A ,  B >. )  =  ( 2nd `  p
) )
1110eqeq1d 2459 . . . . . . 7  |-  ( p  =  <. A ,  B >.  ->  ( ( 2nd `  <. A ,  B >. )  =  B  <->  ( 2nd `  p )  =  B ) )
128, 11syl5ibcom 220 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( p  =  <. A ,  B >.  ->  ( 2nd `  p )  =  B ) )
13123adant3 1016 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  ( p  =  <. A ,  B >.  ->  ( 2nd `  p )  =  B ) )
1413imp 429 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  p  =  <. A ,  B >. )  ->  ( 2nd `  p
)  =  B )
15 op1stg 6811 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
16 fveq2 5872 . . . . . . . . 9  |-  ( p  =  <. A ,  B >.  ->  ( 1st `  p
)  =  ( 1st `  <. A ,  B >. ) )
1716eqcomd 2465 . . . . . . . 8  |-  ( p  =  <. A ,  B >.  ->  ( 1st `  <. A ,  B >. )  =  ( 1st `  p
) )
1817eqeq1d 2459 . . . . . . 7  |-  ( p  =  <. A ,  B >.  ->  ( ( 1st `  <. A ,  B >. )  =  A  <->  ( 1st `  p )  =  A ) )
1915, 18syl5ibcom 220 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( p  =  <. A ,  B >.  ->  ( 1st `  p )  =  A ) )
20193adant3 1016 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  ( p  =  <. A ,  B >.  ->  ( 1st `  p )  =  A ) )
2120imp 429 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  p  =  <. A ,  B >. )  ->  ( 1st `  p
)  =  A )
22 simp1 996 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  A  e.  V )
23 simpl2 1000 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  x  =  A )  ->  B  e.  W )
24 mpt2sn.a . . . . . . . . . 10  |-  ( x  =  A  ->  C  =  D )
2524adantl 466 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  x  =  A )  ->  C  =  D )
26 mpt2sn.b . . . . . . . . 9  |-  ( y  =  B  ->  D  =  E )
2725, 26sylan9eq 2518 . . . . . . . 8  |-  ( ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  /\  x  =  A )  /\  y  =  B )  ->  C  =  E )
2823, 27csbied 3457 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  x  =  A )  ->  [_ B  /  y ]_ C  =  E )
2922, 28csbied 3457 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  E
)
3029adantr 465 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  p  =  <. A ,  B >. )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  E
)
31 csbeq1 3433 . . . . . . . 8  |-  ( ( 1st `  p )  =  A  ->  [_ ( 1st `  p )  /  x ]_ [_ ( 2nd `  p )  /  y ]_ C  =  [_ A  /  x ]_ [_ ( 2nd `  p )  / 
y ]_ C )
3231eqeq1d 2459 . . . . . . 7  |-  ( ( 1st `  p )  =  A  ->  ( [_ ( 1st `  p
)  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  E  <->  [_ A  /  x ]_ [_ ( 2nd `  p )  /  y ]_ C  =  E
) )
3332adantl 466 . . . . . 6  |-  ( ( ( 2nd `  p
)  =  B  /\  ( 1st `  p )  =  A )  -> 
( [_ ( 1st `  p
)  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  E  <->  [_ A  /  x ]_ [_ ( 2nd `  p )  /  y ]_ C  =  E
) )
34 csbeq1 3433 . . . . . . . . 9  |-  ( ( 2nd `  p )  =  B  ->  [_ ( 2nd `  p )  / 
y ]_ C  =  [_ B  /  y ]_ C
)
3534adantr 465 . . . . . . . 8  |-  ( ( ( 2nd `  p
)  =  B  /\  ( 1st `  p )  =  A )  ->  [_ ( 2nd `  p
)  /  y ]_ C  =  [_ B  / 
y ]_ C )
3635csbeq2dv 3843 . . . . . . 7  |-  ( ( ( 2nd `  p
)  =  B  /\  ( 1st `  p )  =  A )  ->  [_ A  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  [_ A  /  x ]_ [_ B  / 
y ]_ C )
3736eqeq1d 2459 . . . . . 6  |-  ( ( ( 2nd `  p
)  =  B  /\  ( 1st `  p )  =  A )  -> 
( [_ A  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  E  <->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  E ) )
3833, 37bitrd 253 . . . . 5  |-  ( ( ( 2nd `  p
)  =  B  /\  ( 1st `  p )  =  A )  -> 
( [_ ( 1st `  p
)  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  E  <->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  E ) )
3930, 38syl5ibrcom 222 . . . 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 679 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U
)  /\  p  =  <. A ,  B >. )  ->  [_ ( 1st `  p
)  /  x ]_ [_ ( 2nd `  p
)  /  y ]_ C  =  E )
41 opex 4720 . . . 4  |-  <. A ,  B >.  e.  _V
4241a1i 11 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  -> 
<. A ,  B >.  e. 
_V )
43 simp3 998 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  E  e.  U )
4440, 42, 43fmptsnd 6094 . 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 2508 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 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   [_csb 3430   {csn 4032   <.cop 4038    |-> cmpt 4515    X. cxp 5006   ` cfv 5594    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800
This theorem is referenced by:  mat1dim0  19193  mat1dimid  19194  mat1dimmul  19196  d1mat2pmat  19458
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