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.

Step	Hyp	Ref	Expression
1		n0 3794	. . . . . 6
2	1	biimpi 194	. . . . 5
3	2	biantrurd 508	. . . 4
4		ancom 450	. . . . . . . 8
5	4	anbi1i 695	. . . . . . 7
6		brxp 5035	. . . . . . . 8
7		brxp 5035	. . . . . . . 8
8	6, 7	anbi12i 697	. . . . . . 7
9		anandi 828	. . . . . . 7
10	5, 8, 9	3bitr4i 277	. . . . . 6
11	10	exbii 1667	. . . . 5
12		19.41v 1771	. . . . 5
13	11, 12	bitr2i 250	. . . 4
14	3, 13	syl6rbb 262	. . 3
15	14	opabbidv 4515	. 2
16		df-co 5013	. 2
17		df-xp 5010	. 2
18	15, 16, 17	3eqtr4g 2523	1

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