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Theorem xpco 5552
 Description: Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.)
Assertion
Ref Expression
xpco

Proof of Theorem xpco
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3794 . . . . . 6
21biimpi 194 . . . . 5
32biantrurd 508 . . . 4
4 ancom 450 . . . . . . . 8
54anbi1i 695 . . . . . . 7
6 brxp 5035 . . . . . . . 8
7 brxp 5035 . . . . . . . 8
86, 7anbi12i 697 . . . . . . 7
9 anandi 828 . . . . . . 7
105, 8, 93bitr4i 277 . . . . . 6
1110exbii 1667 . . . . 5
12 19.41v 1771 . . . . 5
1311, 12bitr2i 250 . . . 4
143, 13syl6rbb 262 . . 3
1514opabbidv 4515 . 2
16 df-co 5013 . 2
17 df-xp 5010 . 2
1815, 16, 173eqtr4g 2523 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652   c0 3784   class class class wbr 4452  {copab 4509  X.cxp 5002  o.ccom 5008 This theorem is referenced by:  xpcoid  5553  ustund  20724  ustneism  20726 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-co 5013
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