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Theorem fmptsnd 6073
Description: Express a singleton function in maps-to notation. Deduction form of fmptsng 6072. (Contributed by AV, 4-Aug-2019.)
Hypotheses
Ref Expression
fmptsnd.1  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
fmptsnd.2  |-  ( ph  ->  A  e.  V )
fmptsnd.3  |-  ( ph  ->  C  e.  W )
Assertion
Ref Expression
fmptsnd  |-  ( ph  ->  { <. A ,  C >. }  =  ( x  e.  { A }  |->  B ) )
Distinct variable groups:    x, A    x, C    ph, x
Allowed substitution hints:    B( x)    V( x)    W( x)

Proof of Theorem fmptsnd
Dummy variables  p  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsn 3986 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
21bicomi 202 . . . 4  |-  ( x  =  A  <->  x  e.  { A } )
32anbi1i 693 . . 3  |-  ( ( x  =  A  /\  y  =  B )  <->  ( x  e.  { A }  /\  y  =  B ) )
43opabbii 4459 . 2  |-  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) }  =  { <. x ,  y >.  |  ( x  e.  { A }  /\  y  =  B ) }
5 elsn 3986 . . . . 5  |-  ( p  e.  { <. A ,  C >. }  <->  p  =  <. A ,  C >. )
6 eqidd 2403 . . . . . . . 8  |-  ( ph  ->  A  =  A )
7 eqidd 2403 . . . . . . . 8  |-  ( ph  ->  C  =  C )
8 sbcan 3320 . . . . . . . . . . 11  |-  ( [. C  /  y ]. (
x  =  A  /\  y  =  B )  <->  (
[. C  /  y ]. x  =  A  /\  [. C  /  y ]. y  =  B
) )
9 fmptsnd.3 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  W )
10 sbcg 3343 . . . . . . . . . . . . 13  |-  ( C  e.  W  ->  ( [. C  /  y ]. x  =  A  <->  x  =  A ) )
119, 10syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( [. C  / 
y ]. x  =  A  <-> 
x  =  A ) )
12 eqsbc3 3317 . . . . . . . . . . . . 13  |-  ( C  e.  W  ->  ( [. C  /  y ]. y  =  B  <->  C  =  B ) )
139, 12syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( [. C  / 
y ]. y  =  B  <-> 
C  =  B ) )
1411, 13anbi12d 709 . . . . . . . . . . 11  |-  ( ph  ->  ( ( [. C  /  y ]. x  =  A  /\  [. C  /  y ]. y  =  B )  <->  ( x  =  A  /\  C  =  B ) ) )
158, 14syl5bb 257 . . . . . . . . . 10  |-  ( ph  ->  ( [. C  / 
y ]. ( x  =  A  /\  y  =  B )  <->  ( x  =  A  /\  C  =  B ) ) )
1615sbcbidv 3332 . . . . . . . . 9  |-  ( ph  ->  ( [. A  /  x ]. [. C  / 
y ]. ( x  =  A  /\  y  =  B )  <->  [. A  /  x ]. ( x  =  A  /\  C  =  B ) ) )
17 fmptsnd.2 . . . . . . . . . 10  |-  ( ph  ->  A  e.  V )
18 eqeq1 2406 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  =  A  <->  A  =  A ) )
1918adantl 464 . . . . . . . . . . 11  |-  ( (
ph  /\  x  =  A )  ->  (
x  =  A  <->  A  =  A ) )
20 fmptsnd.1 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
2120eqeq2d 2416 . . . . . . . . . . 11  |-  ( (
ph  /\  x  =  A )  ->  ( C  =  B  <->  C  =  C ) )
2219, 21anbi12d 709 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  A )  ->  (
( x  =  A  /\  C  =  B )  <->  ( A  =  A  /\  C  =  C ) ) )
2317, 22sbcied 3314 . . . . . . . . 9  |-  ( ph  ->  ( [. A  /  x ]. ( x  =  A  /\  C  =  B )  <->  ( A  =  A  /\  C  =  C ) ) )
2416, 23bitrd 253 . . . . . . . 8  |-  ( ph  ->  ( [. A  /  x ]. [. C  / 
y ]. ( x  =  A  /\  y  =  B )  <->  ( A  =  A  /\  C  =  C ) ) )
256, 7, 24mpbir2and 923 . . . . . . 7  |-  ( ph  ->  [. A  /  x ]. [. C  /  y ]. ( x  =  A  /\  y  =  B ) )
26 opelopabsb 4700 . . . . . . 7  |-  ( <. A ,  C >.  e. 
{ <. x ,  y
>.  |  ( x  =  A  /\  y  =  B ) }  <->  [. A  /  x ]. [. C  / 
y ]. ( x  =  A  /\  y  =  B ) )
2725, 26sylibr 212 . . . . . 6  |-  ( ph  -> 
<. A ,  C >.  e. 
{ <. x ,  y
>.  |  ( x  =  A  /\  y  =  B ) } )
28 eleq1 2474 . . . . . 6  |-  ( p  =  <. A ,  C >.  ->  ( p  e. 
{ <. x ,  y
>.  |  ( x  =  A  /\  y  =  B ) }  <->  <. A ,  C >.  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } ) )
2927, 28syl5ibrcom 222 . . . . 5  |-  ( ph  ->  ( p  =  <. A ,  C >.  ->  p  e.  { <. x ,  y
>.  |  ( x  =  A  /\  y  =  B ) } ) )
305, 29syl5bi 217 . . . 4  |-  ( ph  ->  ( p  e.  { <. A ,  C >. }  ->  p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } ) )
31 elopab 4698 . . . . 5  |-  ( p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } 
<->  E. x E. y
( p  =  <. x ,  y >.  /\  (
x  =  A  /\  y  =  B )
) )
32 opeq12 4161 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
3332adantl 464 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  <. x ,  y >.  =  <. A ,  B >. )
3433eqeq2d 2416 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( p  =  <. x ,  y >.  <->  p  =  <. A ,  B >. ) )
3520adantrr 715 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  B  =  C )
3635opeq2d 4166 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  <. A ,  B >.  = 
<. A ,  C >. )
37 opex 4655 . . . . . . . . . . . . 13  |-  <. A ,  C >.  e.  _V
3837snid 4000 . . . . . . . . . . . 12  |-  <. A ,  C >.  e.  { <. A ,  C >. }
3936, 38syl6eqel 2498 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  <. A ,  B >.  e. 
{ <. A ,  C >. } )
40 eleq1 2474 . . . . . . . . . . 11  |-  ( p  =  <. A ,  B >.  ->  ( p  e. 
{ <. A ,  C >. }  <->  <. A ,  B >.  e.  { <. A ,  C >. } ) )
4139, 40syl5ibrcom 222 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( p  =  <. A ,  B >.  ->  p  e.  { <. A ,  C >. } ) )
4234, 41sylbid 215 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( p  =  <. x ,  y >.  ->  p  e.  { <. A ,  C >. } ) )
4342ex 432 . . . . . . . 8  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  (
p  =  <. x ,  y >.  ->  p  e.  { <. A ,  C >. } ) ) )
4443com23 78 . . . . . . 7  |-  ( ph  ->  ( p  =  <. x ,  y >.  ->  (
( x  =  A  /\  y  =  B )  ->  p  e.  {
<. A ,  C >. } ) ) )
4544impd 429 . . . . . 6  |-  ( ph  ->  ( ( p  = 
<. x ,  y >.  /\  ( x  =  A  /\  y  =  B ) )  ->  p  e.  { <. A ,  C >. } ) )
4645exlimdvv 1746 . . . . 5  |-  ( ph  ->  ( E. x E. y ( p  = 
<. x ,  y >.  /\  ( x  =  A  /\  y  =  B ) )  ->  p  e.  { <. A ,  C >. } ) )
4731, 46syl5bi 217 . . . 4  |-  ( ph  ->  ( p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) }  ->  p  e.  { <. A ,  C >. } ) )
4830, 47impbid 190 . . 3  |-  ( ph  ->  ( p  e.  { <. A ,  C >. }  <-> 
p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } ) )
4948eqrdv 2399 . 2  |-  ( ph  ->  { <. A ,  C >. }  =  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } )
50 df-mpt 4455 . . 3  |-  ( x  e.  { A }  |->  B )  =  { <. x ,  y >.  |  ( x  e. 
{ A }  /\  y  =  B ) }
5150a1i 11 . 2  |-  ( ph  ->  ( x  e.  { A }  |->  B )  =  { <. x ,  y >.  |  ( x  e.  { A }  /\  y  =  B ) } )
524, 49, 513eqtr4a 2469 1  |-  ( ph  ->  { <. A ,  C >. }  =  ( x  e.  { A }  |->  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   [.wsbc 3277   {csn 3972   <.cop 3978   {copab 4452    |-> cmpt 4453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-opab 4454  df-mpt 4455
This theorem is referenced by:  fmptapd  6075  fmptpr  6076  mpt2sn  6875
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