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Theorem fmptsng 6091
Description: Express a singleton function in maps-to notation. Version of fmptsn 6090 allowing the mapping value to depend on the mapping variable (usual case). (Contributed by AV, 27-Feb-2019.)
Hypothesis
Ref Expression
fmptsng.1  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
fmptsng  |-  ( ( A  e.  V  /\  C  e.  W )  ->  { <. A ,  C >. }  =  ( x  e.  { A }  |->  B ) )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    B( x)    V( x)    W( x)

Proof of Theorem fmptsng
Dummy variables  p  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsn 4007 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
21bicomi 205 . . . 4  |-  ( x  =  A  <->  x  e.  { A } )
32anbi1i 699 . . 3  |-  ( ( x  =  A  /\  y  =  B )  <->  ( x  e.  { A }  /\  y  =  B ) )
43opabbii 4481 . 2  |-  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) }  =  { <. x ,  y >.  |  ( x  e.  { A }  /\  y  =  B ) }
5 elsn 4007 . . . . 5  |-  ( p  e.  { <. A ,  C >. }  <->  p  =  <. A ,  C >. )
6 eqidd 2421 . . . . . . 7  |-  ( ( A  e.  V  /\  C  e.  W )  ->  A  =  A )
7 eqidd 2421 . . . . . . 7  |-  ( ( A  e.  V  /\  C  e.  W )  ->  C  =  C )
8 eqeq1 2424 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  =  A  <->  A  =  A ) )
98adantr 466 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  C )  ->  ( x  =  A  <-> 
A  =  A ) )
10 eqeq1 2424 . . . . . . . . . 10  |-  ( y  =  C  ->  (
y  =  B  <->  C  =  B ) )
11 fmptsng.1 . . . . . . . . . . 11  |-  ( x  =  A  ->  B  =  C )
1211eqeq2d 2434 . . . . . . . . . 10  |-  ( x  =  A  ->  ( C  =  B  <->  C  =  C ) )
1310, 12sylan9bbr 705 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  C )  ->  ( y  =  B  <-> 
C  =  C ) )
149, 13anbi12d 715 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  C )  ->  ( ( x  =  A  /\  y  =  B )  <->  ( A  =  A  /\  C  =  C ) ) )
1514opelopabga 4725 . . . . . . 7  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( <. A ,  C >.  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } 
<->  ( A  =  A  /\  C  =  C ) ) )
166, 7, 15mpbir2and 930 . . . . . 6  |-  ( ( A  e.  V  /\  C  e.  W )  -> 
<. A ,  C >.  e. 
{ <. x ,  y
>.  |  ( x  =  A  /\  y  =  B ) } )
17 eleq1 2492 . . . . . 6  |-  ( p  =  <. A ,  C >.  ->  ( p  e. 
{ <. x ,  y
>.  |  ( x  =  A  /\  y  =  B ) }  <->  <. A ,  C >.  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } ) )
1816, 17syl5ibrcom 225 . . . . 5  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( p  =  <. A ,  C >.  ->  p  e.  { <. x ,  y
>.  |  ( x  =  A  /\  y  =  B ) } ) )
195, 18syl5bi 220 . . . 4  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( p  e.  { <. A ,  C >. }  ->  p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } ) )
20 elopab 4720 . . . . 5  |-  ( p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } 
<->  E. x E. y
( p  =  <. x ,  y >.  /\  (
x  =  A  /\  y  =  B )
) )
21 opeq12 4183 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
2221eqeq2d 2434 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  ->  ( p  =  <. x ,  y >.  <->  p  =  <. A ,  B >. ) )
2311adantr 466 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  B )  ->  B  =  C )
2423opeq2d 4188 . . . . . . . . . . 11  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. A ,  B >.  = 
<. A ,  C >. )
25 opex 4677 . . . . . . . . . . . 12  |-  <. A ,  C >.  e.  _V
2625snid 4021 . . . . . . . . . . 11  |-  <. A ,  C >.  e.  { <. A ,  C >. }
2724, 26syl6eqel 2516 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. A ,  B >.  e. 
{ <. A ,  C >. } )
28 eleq1 2492 . . . . . . . . . 10  |-  ( p  =  <. A ,  B >.  ->  ( p  e. 
{ <. A ,  C >. }  <->  <. A ,  B >.  e.  { <. A ,  C >. } ) )
2927, 28syl5ibrcom 225 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  ->  ( p  =  <. A ,  B >.  ->  p  e.  { <. A ,  C >. } ) )
3022, 29sylbid 218 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( p  =  <. x ,  y >.  ->  p  e.  { <. A ,  C >. } ) )
3130impcom 431 . . . . . . 7  |-  ( ( p  =  <. x ,  y >.  /\  (
x  =  A  /\  y  =  B )
)  ->  p  e.  {
<. A ,  C >. } )
3231exlimivv 1767 . . . . . 6  |-  ( E. x E. y ( p  =  <. x ,  y >.  /\  (
x  =  A  /\  y  =  B )
)  ->  p  e.  {
<. A ,  C >. } )
3332a1i 11 . . . . 5  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( E. x E. y ( p  = 
<. x ,  y >.  /\  ( x  =  A  /\  y  =  B ) )  ->  p  e.  { <. A ,  C >. } ) )
3420, 33syl5bi 220 . . . 4  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) }  ->  p  e.  { <. A ,  C >. } ) )
3519, 34impbid 193 . . 3  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( p  e.  { <. A ,  C >. }  <-> 
p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } ) )
3635eqrdv 2417 . 2  |-  ( ( A  e.  V  /\  C  e.  W )  ->  { <. A ,  C >. }  =  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } )
37 df-mpt 4477 . . 3  |-  ( x  e.  { A }  |->  B )  =  { <. x ,  y >.  |  ( x  e. 
{ A }  /\  y  =  B ) }
3837a1i 11 . 2  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( x  e.  { A }  |->  B )  =  { <. x ,  y >.  |  ( x  e.  { A }  /\  y  =  B ) } )
394, 36, 383eqtr4a 2487 1  |-  ( ( A  e.  V  /\  C  e.  W )  ->  { <. A ,  C >. }  =  ( x  e.  { A }  |->  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867   {csn 3993   <.cop 3999   {copab 4474    |-> cmpt 4475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-opab 4476  df-mpt 4477
This theorem is referenced by:  mdet0pr  19554  m1detdiag  19559
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