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Theorem xpsnen 7666
Description: A set is equinumerous to its Cartesian 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
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsnen.1 . . 3  |-  A  e. 
_V
2 snex 4662 . . 3  |-  { B }  e.  _V
31, 2xpex 6610 . 2  |-  ( A  X.  { B }
)  e.  _V
4 elxp 4870 . . 3  |-  ( y  e.  ( A  X.  { B } )  <->  E. x E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) ) )
5 inteq 4258 . . . . . . . 8  |-  ( y  =  <. x ,  z
>.  ->  |^| y  =  |^| <.
x ,  z >.
)
65inteqd 4260 . . . . . . 7  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  =  |^| |^|
<. x ,  z >.
)
7 vex 3083 . . . . . . . 8  |-  x  e. 
_V
8 vex 3083 . . . . . . . 8  |-  z  e. 
_V
97, 8op1stb 4691 . . . . . . 7  |-  |^| |^| <. x ,  z >.  =  x
106, 9syl6eq 2479 . . . . . 6  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  =  x )
1110, 7syl6eqel 2515 . . . . 5  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  e.  _V )
1211adantr 466 . . . 4  |-  ( ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  ->  |^| |^| y  e.  _V )
1312exlimivv 1771 . . 3  |-  ( E. x E. z ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  ->  |^| |^| y  e.  _V )
144, 13sylbi 198 . 2  |-  ( y  e.  ( A  X.  { B } )  ->  |^| |^| y  e.  _V )
15 opex 4685 . . 3  |-  <. x ,  B >.  e.  _V
1615a1i 11 . 2  |-  ( x  e.  A  ->  <. x ,  B >.  e.  _V )
17 eqvisset 3088 . . . . 5  |-  ( x  =  |^| |^| y  ->  |^| |^| y  e.  _V )
18 ancom 451 . . . . . . . . . . 11  |-  ( ( ( y  =  <. x ,  z >.  /\  x  e.  A )  /\  z  e.  { B } )  <-> 
( z  e.  { B }  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
19 anass 653 . . . . . . . . . . 11  |-  ( ( ( y  =  <. x ,  z >.  /\  x  e.  A )  /\  z  e.  { B } )  <-> 
( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) ) )
20 elsn 4012 . . . . . . . . . . . 12  |-  ( z  e.  { B }  <->  z  =  B )
2120anbi1i 699 . . . . . . . . . . 11  |-  ( ( z  e.  { B }  /\  ( y  = 
<. x ,  z >.  /\  x  e.  A
) )  <->  ( z  =  B  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
2218, 19, 213bitr3i 278 . . . . . . . . . 10  |-  ( ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  <->  ( z  =  B  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
2322exbii 1712 . . . . . . . . 9  |-  ( E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) )  <->  E. z
( z  =  B  /\  ( y  = 
<. x ,  z >.  /\  x  e.  A
) ) )
24 xpsnen.2 . . . . . . . . . 10  |-  B  e. 
_V
25 opeq2 4188 . . . . . . . . . . . 12  |-  ( z  =  B  ->  <. x ,  z >.  =  <. x ,  B >. )
2625eqeq2d 2436 . . . . . . . . . . 11  |-  ( z  =  B  ->  (
y  =  <. x ,  z >.  <->  y  =  <. x ,  B >. ) )
2726anbi1d 709 . . . . . . . . . 10  |-  ( z  =  B  ->  (
( y  =  <. x ,  z >.  /\  x  e.  A )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
2824, 27ceqsexv 3118 . . . . . . . . 9  |-  ( E. z ( z  =  B  /\  ( y  =  <. x ,  z
>.  /\  x  e.  A
) )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) )
29 inteq 4258 . . . . . . . . . . . . . 14  |-  ( y  =  <. x ,  B >.  ->  |^| y  =  |^| <.
x ,  B >. )
3029inteqd 4260 . . . . . . . . . . . . 13  |-  ( y  =  <. x ,  B >.  ->  |^| |^| y  =  |^| |^|
<. x ,  B >. )
317, 24op1stb 4691 . . . . . . . . . . . . 13  |-  |^| |^| <. x ,  B >.  =  x
3230, 31syl6req 2480 . . . . . . . . . . . 12  |-  ( y  =  <. x ,  B >.  ->  x  =  |^| |^| y )
3332pm4.71ri 637 . . . . . . . . . . 11  |-  ( y  =  <. x ,  B >.  <-> 
( x  =  |^| |^| y  /\  y  = 
<. x ,  B >. ) )
3433anbi1i 699 . . . . . . . . . 10  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( (
x  =  |^| |^| y  /\  y  =  <. x ,  B >. )  /\  x  e.  A
) )
35 anass 653 . . . . . . . . . 10  |-  ( ( ( x  =  |^| |^| y  /\  y  = 
<. x ,  B >. )  /\  x  e.  A
)  <->  ( x  = 
|^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3634, 35bitri 252 . . . . . . . . 9  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( x  =  |^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3723, 28, 363bitri 274 . . . . . . . 8  |-  ( E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) )  <->  ( x  =  |^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3837exbii 1712 . . . . . . 7  |-  ( E. x E. z ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  <->  E. x
( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
394, 38bitri 252 . . . . . 6  |-  ( y  e.  ( A  X.  { B } )  <->  E. x
( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
40 opeq1 4187 . . . . . . . . 9  |-  ( x  =  |^| |^| y  -> 
<. x ,  B >.  = 
<. |^| |^| y ,  B >. )
4140eqeq2d 2436 . . . . . . . 8  |-  ( x  =  |^| |^| y  ->  ( y  =  <. x ,  B >.  <->  y  =  <. |^| |^| y ,  B >. ) )
42 eleq1 2495 . . . . . . . 8  |-  ( x  =  |^| |^| y  ->  ( x  e.  A  <->  |^|
|^| y  e.  A
) )
4341, 42anbi12d 715 . . . . . . 7  |-  ( x  =  |^| |^| y  ->  ( ( y  = 
<. x ,  B >.  /\  x  e.  A )  <-> 
( y  =  <. |^|
|^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4443ceqsexgv 3203 . . . . . 6  |-  ( |^| |^| y  e.  _V  ->  ( E. x ( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A ) )  <->  ( y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4539, 44syl5bb 260 . . . . 5  |-  ( |^| |^| y  e.  _V  ->  ( y  e.  ( A  X.  { B }
)  <->  ( y  = 
<. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4617, 45syl 17 . . . 4  |-  ( x  =  |^| |^| y  ->  ( y  e.  ( A  X.  { B } )  <->  ( y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4746pm5.32ri 642 . . 3  |-  ( ( y  e.  ( A  X.  { B }
)  /\  x  =  |^| |^| y )  <->  ( (
y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A )  /\  x  =  |^| |^| y ) )
4832adantr 466 . . . . 5  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  ->  x  =  |^| |^| y )
4948pm4.71i 636 . . . 4  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( (
y  =  <. x ,  B >.  /\  x  e.  A )  /\  x  =  |^| |^| y ) )
5043pm5.32ri 642 . . . 4  |-  ( ( ( y  =  <. x ,  B >.  /\  x  e.  A )  /\  x  =  |^| |^| y )  <->  ( (
y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A )  /\  x  =  |^| |^| y ) )
5149, 50bitr2i 253 . . 3  |-  ( ( ( y  =  <. |^|
|^| y ,  B >.  /\  |^| |^| y  e.  A
)  /\  x  =  |^| |^| y )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) )
52 ancom 451 . . 3  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( x  e.  A  /\  y  =  <. x ,  B >. ) )
5347, 51, 523bitri 274 . 2  |-  ( ( y  e.  ( A  X.  { B }
)  /\  x  =  |^| |^| y )  <->  ( x  e.  A  /\  y  =  <. x ,  B >. ) )
543, 1, 14, 16, 53en2i 7618 1  |-  ( A  X.  { B }
)  ~~  A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   _Vcvv 3080   {csn 3998   <.cop 4004   |^|cint 4255   class class class wbr 4423    X. cxp 4851    ~~ cen 7578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-int 4256  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-en 7582
This theorem is referenced by:  xpsneng  7667  endisj  7669  infxpenlem  8453  pm110.643  8615  hashxplem  12610  rexpen  14280  heiborlem3  32110
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