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Theorem xpco 5530
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 3793 . . . . . 6  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
21biimpi 194 . . . . 5  |-  ( B  =/=  (/)  ->  E. y 
y  e.  B )
32biantrurd 506 . . . 4  |-  ( B  =/=  (/)  ->  ( (
x  e.  A  /\  z  e.  C )  <->  ( E. y  y  e.  B  /\  ( x  e.  A  /\  z  e.  C ) ) ) )
4 ancom 448 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  <->  ( y  e.  B  /\  x  e.  A )
)
54anbi1i 693 . . . . . . 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 5019 . . . . . . . 8  |-  ( x ( A  X.  B
) y  <->  ( x  e.  A  /\  y  e.  B ) )
7 brxp 5019 . . . . . . . 8  |-  ( y ( B  X.  C
) z  <->  ( y  e.  B  /\  z  e.  C ) )
86, 7anbi12i 695 . . . . . . 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 1672 . . . . 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 1776 . . . . 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 4502 . 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 4997 . 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 4994 . 2  |-  ( A  X.  C )  =  { <. x ,  z
>.  |  ( x  e.  A  /\  z  e.  C ) }
1815, 16, 173eqtr4g 2520 1  |-  ( B  =/=  (/)  ->  ( ( B  X.  C )  o.  ( A  X.  B
) )  =  ( A  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   (/)c0 3783   class class class wbr 4439   {copab 4496    X. cxp 4986    o. ccom 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-co 4997
This theorem is referenced by:  xpcoid  5531  ustund  20893  ustneism  20895
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