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Theorem xpco 5545
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 3794 . . . . . 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 5029 . . . . . . . 8  |-  ( x ( A  X.  B
) y  <->  ( x  e.  A  /\  y  e.  B ) )
7 brxp 5029 . . . . . . . 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 826 . . . . . . 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 1644 . . . . 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 1945 . . . . 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 4510 . 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 5008 . 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 5005 . 2  |-  ( A  X.  C )  =  { <. x ,  z
>.  |  ( x  e.  A  /\  z  e.  C ) }
1815, 16, 173eqtr4g 2533 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 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   (/)c0 3785   class class class wbr 4447   {copab 4504    X. cxp 4997    o. ccom 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-co 5008
This theorem is referenced by:  xpcoid  5546  ustund  20456  ustneism  20458
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