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Theorem fmptsnd 30745
Description: Express a singleton function in maps-to notation. Deduction form of fmptsng 5919. (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 3910 . . . . 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 4375 . 2  |-  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) }  =  { <. x ,  y >.  |  ( x  e.  { A }  /\  y  =  B ) }
5 elsn 3910 . . . . 5  |-  ( p  e.  { <. A ,  C >. }  <->  p  =  <. A ,  C >. )
6 eqidd 2444 . . . . . . . 8  |-  ( ph  ->  A  =  A )
7 eqidd 2444 . . . . . . . 8  |-  ( ph  ->  C  =  C )
8 sbcan 3248 . . . . . . . . . . 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 3279 . . . . . . . . . . . . 13  |-  ( C  e.  W  ->  ( [. C  /  y ]. x  =  A  <->  x  =  A ) )
119, 10syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( [. C  / 
y ]. x  =  A  <-> 
x  =  A ) )
12 eqsbc3 3245 . . . . . . . . . . . . 13  |-  ( C  e.  W  ->  ( [. C  /  y ]. y  =  B  <->  C  =  B ) )
139, 12syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( [. C  / 
y ]. y  =  B  <-> 
C  =  B ) )
1411, 13anbi12d 710 . . . . . . . . . . 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 3264 . . . . . . . . 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 2449 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  =  A  <->  A  =  A ) )
1918adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  x  =  A )  ->  (
x  =  A  <->  A  =  A ) )
20 fmptsnd.1 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
2120eqeq2d 2454 . . . . . . . . . . 11  |-  ( (
ph  /\  x  =  A )  ->  ( C  =  B  <->  C  =  C ) )
2219, 21anbi12d 710 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  A )  ->  (
( x  =  A  /\  C  =  B )  <->  ( A  =  A  /\  C  =  C ) ) )
2317, 22sbcied 3242 . . . . . . . . 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 913 . . . . . . 7  |-  ( ph  ->  [. A  /  x ]. [. C  /  y ]. ( x  =  A  /\  y  =  B ) )
26 opelopabsb 4618 . . . . . . 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 2503 . . . . . 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 4616 . . . . 5  |-  ( p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } 
<->  E. x E. y
( p  =  <. x ,  y >.  /\  (
x  =  A  /\  y  =  B )
) )
32 opeq12 4080 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
3332adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  <. x ,  y >.  =  <. A ,  B >. )
3433eqeq2d 2454 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( p  =  <. x ,  y >.  <->  p  =  <. A ,  B >. ) )
3520adantrr 716 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  B  =  C )
3635opeq2d 4085 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  <. A ,  B >.  = 
<. A ,  C >. )
37 opex 4575 . . . . . . . . . . . . 13  |-  <. A ,  C >.  e.  _V
3837snid 3924 . . . . . . . . . . . 12  |-  <. A ,  C >.  e.  { <. A ,  C >. }
3936, 38syl6eqel 2531 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  <. A ,  B >.  e. 
{ <. A ,  C >. } )
40 eleq1 2503 . . . . . . . . . . 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 434 . . . . . . . 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 431 . . . . . 6  |-  ( ph  ->  ( ( p  = 
<. x ,  y >.  /\  ( x  =  A  /\  y  =  B ) )  ->  p  e.  { <. A ,  C >. } ) )
4645exlimdvv 1691 . . . . 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 191 . . 3  |-  ( ph  ->  ( p  e.  { <. A ,  C >. }  <-> 
p  e.  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } ) )
4948eqrdv 2441 . 2  |-  ( ph  ->  { <. A ,  C >. }  =  { <. x ,  y >.  |  ( x  =  A  /\  y  =  B ) } )
50 df-mpt 4371 . . 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 2501 1  |-  ( ph  ->  { <. A ,  C >. }  =  ( x  e.  { A }  |->  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   [.wsbc 3205   {csn 3896   <.cop 3902   {copab 4368    e. cmpt 4369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-opab 4370  df-mpt 4371
This theorem is referenced by:  mpt2sn  30746
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