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Theorem xpsnen 6708
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 xpsnen.2 . . . 4  |-  B  e. 
_V
32snex 4104 . . 3  |-  { B }  e.  _V
41, 3xpex 4649 . 2  |-  ( A  X.  { B }
)  e.  _V
5 elxp 4551 . . 3  |-  ( y  e.  ( A  X.  { B } )  <->  E. x E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) ) )
6 inteq 3769 . . . . . . . 8  |-  ( y  =  <. x ,  z
>.  ->  |^| y  =  |^| <.
x ,  z >.
)
76inteqd 3771 . . . . . . 7  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  =  |^| |^|
<. x ,  z >.
)
8 vex 2684 . . . . . . . 8  |-  x  e. 
_V
9 vex 2684 . . . . . . . 8  |-  z  e. 
_V
108, 9op1stb 4394 . . . . . . 7  |-  |^| |^| <. x ,  z >.  =  x
117, 10syl6eq 2186 . . . . . 6  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  =  x )
1211, 8eqeltrdi 2228 . . . . 5  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  e.  _V )
1312adantr 274 . . . 4  |-  ( ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  ->  |^| |^| y  e.  _V )
1413exlimivv 1868 . . 3  |-  ( E. x E. z ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  ->  |^| |^| y  e.  _V )
155, 14sylbi 120 . 2  |-  ( y  e.  ( A  X.  { B } )  ->  |^| |^| y  e.  _V )
168, 2opex 4146 . . 3  |-  <. x ,  B >.  e.  _V
1716a1i 9 . 2  |-  ( x  e.  A  ->  <. x ,  B >.  e.  _V )
18 eqvisset 2691 . . . . 5  |-  ( x  =  |^| |^| y  ->  |^| |^| y  e.  _V )
19 ancom 264 . . . . . . . . . . 11  |-  ( ( ( y  =  <. x ,  z >.  /\  x  e.  A )  /\  z  e.  { B } )  <-> 
( z  e.  { B }  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
20 anass 398 . . . . . . . . . . 11  |-  ( ( ( y  =  <. x ,  z >.  /\  x  e.  A )  /\  z  e.  { B } )  <-> 
( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) ) )
21 velsn 3539 . . . . . . . . . . . 12  |-  ( z  e.  { B }  <->  z  =  B )
2221anbi1i 453 . . . . . . . . . . 11  |-  ( ( z  e.  { B }  /\  ( y  = 
<. x ,  z >.  /\  x  e.  A
) )  <->  ( z  =  B  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
2319, 20, 223bitr3i 209 . . . . . . . . . 10  |-  ( ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  <->  ( z  =  B  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
2423exbii 1584 . . . . . . . . 9  |-  ( E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) )  <->  E. z
( z  =  B  /\  ( y  = 
<. x ,  z >.  /\  x  e.  A
) ) )
25 opeq2 3701 . . . . . . . . . . . 12  |-  ( z  =  B  ->  <. x ,  z >.  =  <. x ,  B >. )
2625eqeq2d 2149 . . . . . . . . . . 11  |-  ( z  =  B  ->  (
y  =  <. x ,  z >.  <->  y  =  <. x ,  B >. ) )
2726anbi1d 460 . . . . . . . . . 10  |-  ( z  =  B  ->  (
( y  =  <. x ,  z >.  /\  x  e.  A )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
282, 27ceqsexv 2720 . . . . . . . . 9  |-  ( E. z ( z  =  B  /\  ( y  =  <. x ,  z
>.  /\  x  e.  A
) )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) )
29 inteq 3769 . . . . . . . . . . . . . 14  |-  ( y  =  <. x ,  B >.  ->  |^| y  =  |^| <.
x ,  B >. )
3029inteqd 3771 . . . . . . . . . . . . 13  |-  ( y  =  <. x ,  B >.  ->  |^| |^| y  =  |^| |^|
<. x ,  B >. )
318, 2op1stb 4394 . . . . . . . . . . . . 13  |-  |^| |^| <. x ,  B >.  =  x
3230, 31syl6req 2187 . . . . . . . . . . . 12  |-  ( y  =  <. x ,  B >.  ->  x  =  |^| |^| y )
3332pm4.71ri 389 . . . . . . . . . . 11  |-  ( y  =  <. x ,  B >.  <-> 
( x  =  |^| |^| y  /\  y  = 
<. x ,  B >. ) )
3433anbi1i 453 . . . . . . . . . 10  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( (
x  =  |^| |^| y  /\  y  =  <. x ,  B >. )  /\  x  e.  A
) )
35 anass 398 . . . . . . . . . 10  |-  ( ( ( x  =  |^| |^| y  /\  y  = 
<. x ,  B >. )  /\  x  e.  A
)  <->  ( x  = 
|^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3634, 35bitri 183 . . . . . . . . 9  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( x  =  |^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3724, 28, 363bitri 205 . . . . . . . 8  |-  ( E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) )  <->  ( x  =  |^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3837exbii 1584 . . . . . . 7  |-  ( E. x E. z ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  <->  E. x
( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
395, 38bitri 183 . . . . . 6  |-  ( y  e.  ( A  X.  { B } )  <->  E. x
( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
40 opeq1 3700 . . . . . . . . 9  |-  ( x  =  |^| |^| y  -> 
<. x ,  B >.  = 
<. |^| |^| y ,  B >. )
4140eqeq2d 2149 . . . . . . . 8  |-  ( x  =  |^| |^| y  ->  ( y  =  <. x ,  B >.  <->  y  =  <. |^| |^| y ,  B >. ) )
42 eleq1 2200 . . . . . . . 8  |-  ( x  =  |^| |^| y  ->  ( x  e.  A  <->  |^|
|^| y  e.  A
) )
4341, 42anbi12d 464 . . . . . . 7  |-  ( x  =  |^| |^| y  ->  ( ( y  = 
<. x ,  B >.  /\  x  e.  A )  <-> 
( y  =  <. |^|
|^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4443ceqsexgv 2809 . . . . . 6  |-  ( |^| |^| y  e.  _V  ->  ( E. x ( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A ) )  <->  ( y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4539, 44syl5bb 191 . . . . 5  |-  ( |^| |^| y  e.  _V  ->  ( y  e.  ( A  X.  { B }
)  <->  ( y  = 
<. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4618, 45syl 14 . . . 4  |-  ( x  =  |^| |^| y  ->  ( y  e.  ( A  X.  { B } )  <->  ( y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4746pm5.32ri 450 . . 3  |-  ( ( y  e.  ( A  X.  { B }
)  /\  x  =  |^| |^| y )  <->  ( (
y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A )  /\  x  =  |^| |^| y ) )
4832adantr 274 . . . . 5  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  ->  x  =  |^| |^| y )
4948pm4.71i 388 . . . 4  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( (
y  =  <. x ,  B >.  /\  x  e.  A )  /\  x  =  |^| |^| y ) )
5043pm5.32ri 450 . . . 4  |-  ( ( ( y  =  <. x ,  B >.  /\  x  e.  A )  /\  x  =  |^| |^| y )  <->  ( (
y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A )  /\  x  =  |^| |^| y ) )
5149, 50bitr2i 184 . . 3  |-  ( ( ( y  =  <. |^|
|^| y ,  B >.  /\  |^| |^| y  e.  A
)  /\  x  =  |^| |^| y )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) )
52 ancom 264 . . 3  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( x  e.  A  /\  y  =  <. x ,  B >. ) )
5347, 51, 523bitri 205 . 2  |-  ( ( y  e.  ( A  X.  { B }
)  /\  x  =  |^| |^| y )  <->  ( x  e.  A  /\  y  =  <. x ,  B >. ) )
544, 1, 15, 17, 53en2i 6657 1  |-  ( A  X.  { B }
)  ~~  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1331   E.wex 1468    e. wcel 1480   _Vcvv 2681   {csn 3522   <.cop 3525   |^|cint 3766   class class class wbr 3924    X. cxp 4532    ~~ cen 6625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-en 6628
This theorem is referenced by:  xpsneng  6709  endisj  6711
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