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Theorem xpco 5461
Description: Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.)
Assertion
Ref Expression
xpco  |-  ( B  =/=  (/)  ->  ( ( B  X.  C )  o.  ( A  X.  B
) )  =  ( A  X.  C ) )

Proof of Theorem xpco
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3730 . . . . . 6  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
21biimpi 194 . . . . 5  |-  ( B  =/=  (/)  ->  E. y 
y  e.  B )
32biantrurd 508 . . . 4  |-  ( B  =/=  (/)  ->  ( (
x  e.  A  /\  z  e.  C )  <->  ( E. y  y  e.  B  /\  ( x  e.  A  /\  z  e.  C ) ) ) )
4 ancom 450 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  <->  ( y  e.  B  /\  x  e.  A )
)
54anbi1i 695 . . . . . . 7  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( y  e.  B  /\  z  e.  C ) )  <->  ( (
y  e.  B  /\  x  e.  A )  /\  ( y  e.  B  /\  z  e.  C
) ) )
6 brxp 4954 . . . . . . . 8  |-  ( x ( A  X.  B
) y  <->  ( x  e.  A  /\  y  e.  B ) )
7 brxp 4954 . . . . . . . 8  |-  ( y ( B  X.  C
) z  <->  ( y  e.  B  /\  z  e.  C ) )
86, 7anbi12i 697 . . . . . . 7  |-  ( ( x ( A  X.  B ) y  /\  y ( B  X.  C ) z )  <-> 
( ( x  e.  A  /\  y  e.  B )  /\  (
y  e.  B  /\  z  e.  C )
) )
9 anandi 824 . . . . . . 7  |-  ( ( y  e.  B  /\  ( x  e.  A  /\  z  e.  C
) )  <->  ( (
y  e.  B  /\  x  e.  A )  /\  ( y  e.  B  /\  z  e.  C
) ) )
105, 8, 93bitr4i 277 . . . . . 6  |-  ( ( x ( A  X.  B ) y  /\  y ( B  X.  C ) z )  <-> 
( y  e.  B  /\  ( x  e.  A  /\  z  e.  C
) ) )
1110exbii 1635 . . . . 5  |-  ( E. y ( x ( A  X.  B ) y  /\  y ( B  X.  C ) z )  <->  E. y
( y  e.  B  /\  ( x  e.  A  /\  z  e.  C
) ) )
12 19.41v 1921 . . . . 5  |-  ( E. y ( y  e.  B  /\  ( x  e.  A  /\  z  e.  C ) )  <->  ( E. y  y  e.  B  /\  ( x  e.  A  /\  z  e.  C
) ) )
1311, 12bitr2i 250 . . . 4  |-  ( ( E. y  y  e.  B  /\  ( x  e.  A  /\  z  e.  C ) )  <->  E. y
( x ( A  X.  B ) y  /\  y ( B  X.  C ) z ) )
143, 13syl6rbb 262 . . 3  |-  ( B  =/=  (/)  ->  ( E. y ( x ( A  X.  B ) y  /\  y ( B  X.  C ) z )  <->  ( x  e.  A  /\  z  e.  C ) ) )
1514opabbidv 4439 . 2  |-  ( B  =/=  (/)  ->  { <. x ,  z >.  |  E. y ( x ( A  X.  B ) y  /\  y ( B  X.  C ) z ) }  =  { <. x ,  z
>.  |  ( x  e.  A  /\  z  e.  C ) } )
16 df-co 4933 . 2  |-  ( ( B  X.  C )  o.  ( A  X.  B ) )  =  { <. x ,  z
>.  |  E. y
( x ( A  X.  B ) y  /\  y ( B  X.  C ) z ) }
17 df-xp 4930 . 2  |-  ( A  X.  C )  =  { <. x ,  z
>.  |  ( x  e.  A  /\  z  e.  C ) }
1815, 16, 173eqtr4g 2515 1  |-  ( B  =/=  (/)  ->  ( ( B  X.  C )  o.  ( A  X.  B
) )  =  ( A  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1757    =/= wne 2641   (/)c0 3721   class class class wbr 4376   {copab 4433    X. cxp 4922    o. ccom 4928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-br 4377  df-opab 4435  df-xp 4930  df-co 4933
This theorem is referenced by:  xpcoid  5462  ustund  19898  ustneism  19900
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