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Theorem xpsnen 6900
Description: A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
xpsnen.1  |-  A  e. 
_V
xpsnen.2  |-  B  e. 
_V
Assertion
Ref Expression
xpsnen  |-  ( A  X.  { B }
)  ~~  A

Proof of Theorem xpsnen
StepHypRef Expression
1 xpsnen.1 . . 3  |-  A  e. 
_V
2 snex 4174 . . 3  |-  { B }  e.  _V
31, 2xpex 4775 . 2  |-  ( A  X.  { B }
)  e.  _V
4 elxp 4680 . . 3  |-  ( y  e.  ( A  X.  { B } )  <->  E. x E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) ) )
5 inteq 3825 . . . . . . . 8  |-  ( y  =  <. x ,  z
>.  ->  |^| y  =  |^| <.
x ,  z >.
)
65inteqd 3827 . . . . . . 7  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  =  |^| |^|
<. x ,  z >.
)
7 vex 2760 . . . . . . . 8  |-  x  e. 
_V
8 vex 2760 . . . . . . . 8  |-  z  e. 
_V
97, 8op1stb 4527 . . . . . . 7  |-  |^| |^| <. x ,  z >.  =  x
106, 9syl6eq 2304 . . . . . 6  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  =  x )
1110, 7syl6eqel 2344 . . . . 5  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  e.  _V )
1211adantr 453 . . . 4  |-  ( ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  ->  |^| |^| y  e.  _V )
1312exlimivv 2026 . . 3  |-  ( E. x E. z ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  ->  |^| |^| y  e.  _V )
144, 13sylbi 189 . 2  |-  ( y  e.  ( A  X.  { B } )  ->  |^| |^| y  e.  _V )
15 opex 4195 . . 3  |-  <. x ,  B >.  e.  _V
1615a1i 12 . 2  |-  ( x  e.  A  ->  <. x ,  B >.  e.  _V )
17 eleq1 2316 . . . . . 6  |-  ( x  =  |^| |^| y  ->  ( x  e.  _V  <->  |^|
|^| y  e.  _V ) )
187, 17mpbii 204 . . . . 5  |-  ( x  =  |^| |^| y  ->  |^| |^| y  e.  _V )
19 ancom 439 . . . . . . . . . . 11  |-  ( ( ( y  =  <. x ,  z >.  /\  x  e.  A )  /\  z  e.  { B } )  <-> 
( z  e.  { B }  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
20 anass 633 . . . . . . . . . . 11  |-  ( ( ( y  =  <. x ,  z >.  /\  x  e.  A )  /\  z  e.  { B } )  <-> 
( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) ) )
21 elsn 3615 . . . . . . . . . . . 12  |-  ( z  e.  { B }  <->  z  =  B )
2221anbi1i 679 . . . . . . . . . . 11  |-  ( ( z  e.  { B }  /\  ( y  = 
<. x ,  z >.  /\  x  e.  A
) )  <->  ( z  =  B  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
2319, 20, 223bitr3i 268 . . . . . . . . . 10  |-  ( ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  <->  ( z  =  B  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
2423exbii 1580 . . . . . . . . 9  |-  ( E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) )  <->  E. z
( z  =  B  /\  ( y  = 
<. x ,  z >.  /\  x  e.  A
) ) )
25 xpsnen.2 . . . . . . . . . 10  |-  B  e. 
_V
26 opeq2 3757 . . . . . . . . . . . 12  |-  ( z  =  B  ->  <. x ,  z >.  =  <. x ,  B >. )
2726eqeq2d 2267 . . . . . . . . . . 11  |-  ( z  =  B  ->  (
y  =  <. x ,  z >.  <->  y  =  <. x ,  B >. ) )
2827anbi1d 688 . . . . . . . . . 10  |-  ( z  =  B  ->  (
( y  =  <. x ,  z >.  /\  x  e.  A )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
2925, 28ceqsexv 2791 . . . . . . . . 9  |-  ( E. z ( z  =  B  /\  ( y  =  <. x ,  z
>.  /\  x  e.  A
) )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) )
30 inteq 3825 . . . . . . . . . . . . . 14  |-  ( y  =  <. x ,  B >.  ->  |^| y  =  |^| <.
x ,  B >. )
3130inteqd 3827 . . . . . . . . . . . . 13  |-  ( y  =  <. x ,  B >.  ->  |^| |^| y  =  |^| |^|
<. x ,  B >. )
327, 25op1stb 4527 . . . . . . . . . . . . 13  |-  |^| |^| <. x ,  B >.  =  x
3331, 32syl6req 2305 . . . . . . . . . . . 12  |-  ( y  =  <. x ,  B >.  ->  x  =  |^| |^| y )
3433pm4.71ri 617 . . . . . . . . . . 11  |-  ( y  =  <. x ,  B >.  <-> 
( x  =  |^| |^| y  /\  y  = 
<. x ,  B >. ) )
3534anbi1i 679 . . . . . . . . . 10  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( (
x  =  |^| |^| y  /\  y  =  <. x ,  B >. )  /\  x  e.  A
) )
36 anass 633 . . . . . . . . . 10  |-  ( ( ( x  =  |^| |^| y  /\  y  = 
<. x ,  B >. )  /\  x  e.  A
)  <->  ( x  = 
|^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3735, 36bitri 242 . . . . . . . . 9  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( x  =  |^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3824, 29, 373bitri 264 . . . . . . . 8  |-  ( E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) )  <->  ( x  =  |^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3938exbii 1580 . . . . . . 7  |-  ( E. x E. z ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  <->  E. x
( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
404, 39bitri 242 . . . . . 6  |-  ( y  e.  ( A  X.  { B } )  <->  E. x
( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
41 opeq1 3756 . . . . . . . . 9  |-  ( x  =  |^| |^| y  -> 
<. x ,  B >.  = 
<. |^| |^| y ,  B >. )
4241eqeq2d 2267 . . . . . . . 8  |-  ( x  =  |^| |^| y  ->  ( y  =  <. x ,  B >.  <->  y  =  <. |^| |^| y ,  B >. ) )
43 eleq1 2316 . . . . . . . 8  |-  ( x  =  |^| |^| y  ->  ( x  e.  A  <->  |^|
|^| y  e.  A
) )
4442, 43anbi12d 694 . . . . . . 7  |-  ( x  =  |^| |^| y  ->  ( ( y  = 
<. x ,  B >.  /\  x  e.  A )  <-> 
( y  =  <. |^|
|^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4544ceqsexgv 2868 . . . . . 6  |-  ( |^| |^| y  e.  _V  ->  ( E. x ( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A ) )  <->  ( y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4640, 45syl5bb 250 . . . . 5  |-  ( |^| |^| y  e.  _V  ->  ( y  e.  ( A  X.  { B }
)  <->  ( y  = 
<. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4718, 46syl 17 . . . 4  |-  ( x  =  |^| |^| y  ->  ( y  e.  ( A  X.  { B } )  <->  ( y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4847pm5.32ri 622 . . 3  |-  ( ( y  e.  ( A  X.  { B }
)  /\  x  =  |^| |^| y )  <->  ( (
y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A )  /\  x  =  |^| |^| y ) )
4933adantr 453 . . . . 5  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  ->  x  =  |^| |^| y )
5049pm4.71i 616 . . . 4  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( (
y  =  <. x ,  B >.  /\  x  e.  A )  /\  x  =  |^| |^| y ) )
5144pm5.32ri 622 . . . 4  |-  ( ( ( y  =  <. x ,  B >.  /\  x  e.  A )  /\  x  =  |^| |^| y )  <->  ( (
y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A )  /\  x  =  |^| |^| y ) )
5250, 51bitr2i 243 . . 3  |-  ( ( ( y  =  <. |^|
|^| y ,  B >.  /\  |^| |^| y  e.  A
)  /\  x  =  |^| |^| y )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) )
53 ancom 439 . . 3  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( x  e.  A  /\  y  =  <. x ,  B >. ) )
5448, 52, 533bitri 264 . 2  |-  ( ( y  e.  ( A  X.  { B }
)  /\  x  =  |^| |^| y )  <->  ( x  e.  A  /\  y  =  <. x ,  B >. ) )
553, 1, 14, 16, 54en2i 6853 1  |-  ( A  X.  { B }
)  ~~  A
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2757   {csn 3600   <.cop 3603   |^|cint 3822   class class class wbr 3983    X. cxp 4645    ~~ cen 6814
This theorem is referenced by:  xpsneng  6901  endisj  6903  infxpenlem  7595  pm110.643  7757  hashxplem  11336  xpnnenOLD  12436  rexpen  12454  heiborlem3  25890
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-int 3823  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-en 6818
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