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Theorem fmptsng 6083
Description: Express a singleton function in maps-to notation. Version of fmptsn 6082 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 4041 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
21bicomi 202 . . . 4  |-  ( x  =  A  <->  x  e.  { A } )
32anbi1i 695 . . 3  |-  ( ( x  =  A  /\  y  =  B )  <->  ( x  e.  { A }  /\  y  =  B ) )
43opabbii 4511 . 2  |-  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) }  =  { <. x ,  y >.  |  ( x  e.  { A }  /\  y  =  B ) }
5 elsn 4041 . . . . 5  |-  ( p  e.  { <. A ,  C >. }  <->  p  =  <. A ,  C >. )
6 eqidd 2468 . . . . . . 7  |-  ( ( A  e.  V  /\  C  e.  W )  ->  A  =  A )
7 eqidd 2468 . . . . . . 7  |-  ( ( A  e.  V  /\  C  e.  W )  ->  C  =  C )
8 eqeq1 2471 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  =  A  <->  A  =  A ) )
98adantr 465 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  C )  ->  ( x  =  A  <-> 
A  =  A ) )
10 eqeq1 2471 . . . . . . . . . 10  |-  ( y  =  C  ->  (
y  =  B  <->  C  =  B ) )
11 fmptsng.1 . . . . . . . . . . 11  |-  ( x  =  A  ->  B  =  C )
1211eqeq2d 2481 . . . . . . . . . 10  |-  ( x  =  A  ->  ( C  =  B  <->  C  =  C ) )
1310, 12sylan9bbr 700 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  C )  ->  ( y  =  B  <-> 
C  =  C ) )
149, 13anbi12d 710 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  C )  ->  ( ( x  =  A  /\  y  =  B )  <->  ( A  =  A  /\  C  =  C ) ) )
1514opelopabga 4760 . . . . . . 7  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( <. A ,  C >.  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } 
<->  ( A  =  A  /\  C  =  C ) ) )
166, 7, 15mpbir2and 920 . . . . . 6  |-  ( ( A  e.  V  /\  C  e.  W )  -> 
<. A ,  C >.  e. 
{ <. x ,  y
>.  |  ( x  =  A  /\  y  =  B ) } )
17 eleq1 2539 . . . . . 6  |-  ( p  =  <. A ,  C >.  ->  ( p  e. 
{ <. x ,  y
>.  |  ( x  =  A  /\  y  =  B ) }  <->  <. A ,  C >.  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } ) )
1816, 17syl5ibrcom 222 . . . . 5  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( p  =  <. A ,  C >.  ->  p  e.  { <. x ,  y
>.  |  ( x  =  A  /\  y  =  B ) } ) )
195, 18syl5bi 217 . . . 4  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( p  e.  { <. A ,  C >. }  ->  p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } ) )
20 elopab 4755 . . . . 5  |-  ( p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } 
<->  E. x E. y
( p  =  <. x ,  y >.  /\  (
x  =  A  /\  y  =  B )
) )
21 opeq12 4215 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
2221eqeq2d 2481 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  ->  ( p  =  <. x ,  y >.  <->  p  =  <. A ,  B >. ) )
2311adantr 465 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  B )  ->  B  =  C )
2423opeq2d 4220 . . . . . . . . . . 11  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. A ,  B >.  = 
<. A ,  C >. )
25 opex 4711 . . . . . . . . . . . 12  |-  <. A ,  C >.  e.  _V
2625snid 4055 . . . . . . . . . . 11  |-  <. A ,  C >.  e.  { <. A ,  C >. }
2724, 26syl6eqel 2563 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. A ,  B >.  e. 
{ <. A ,  C >. } )
28 eleq1 2539 . . . . . . . . . 10  |-  ( p  =  <. A ,  B >.  ->  ( p  e. 
{ <. A ,  C >. }  <->  <. A ,  B >.  e.  { <. A ,  C >. } ) )
2927, 28syl5ibrcom 222 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  ->  ( p  =  <. A ,  B >.  ->  p  e.  { <. A ,  C >. } ) )
3022, 29sylbid 215 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( p  =  <. x ,  y >.  ->  p  e.  { <. A ,  C >. } ) )
3130impcom 430 . . . . . . 7  |-  ( ( p  =  <. x ,  y >.  /\  (
x  =  A  /\  y  =  B )
)  ->  p  e.  {
<. A ,  C >. } )
3231exlimivv 1699 . . . . . 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 217 . . . 4  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) }  ->  p  e.  { <. A ,  C >. } ) )
3519, 34impbid 191 . . 3  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( p  e.  { <. A ,  C >. }  <-> 
p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } ) )
3635eqrdv 2464 . 2  |-  ( ( A  e.  V  /\  C  e.  W )  ->  { <. A ,  C >. }  =  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } )
37 df-mpt 4507 . . 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 2534 1  |-  ( ( A  e.  V  /\  C  e.  W )  ->  { <. A ,  C >. }  =  ( x  e.  { A }  |->  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {csn 4027   <.cop 4033   {copab 4504    |-> cmpt 4505
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
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-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-mpt 4507
This theorem is referenced by:  mdet0pr  18901  m1detdiag  18906
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