Theorem co01 2664

Description: Composition with the empty set.

Assertion

Ref	Expression
co01	|- ((/) o. A) = (/)

Proof of Theorem co01

Step	Hyp	Ref	Expression
1		cnv0 2633	. . . . . 6 |- `'(/) = (/)
2	1	coeq2i 2505	. . . . 5 |- (`'A o. `'(/)) = (`'A o. (/))
3		co02 2663	. . . . 5 |- (`'A o. (/)) = (/)
4	2, 3	eqtr2 1120	. . . 4 |- (/) = (`'A o. `'(/))
5		cnvco 2520	. . . 4 |- `'((/) o. A) = (`'A o. `'(/))
6	4, 1, 5	3eqtr4 1126	. . 3 |- `'(/) = `'((/) o. A)
7		cnveq 2513	. . 3 |- (`'(/) = `'((/) o. A) -> `'`'(/) = `'`'((/) o. A))
8	6, 7	ax-mp 6	. 2 |- `'`'(/) = `'`'((/) o. A)
9		rel0 2499	. . 3 |- Rel (/)
10		dfrel2 2660	. . 3 |- (Rel (/) <-> `'`'(/) = (/))
11	9, 10	mpbi 164	. 2 |- `'`'(/) = (/)
12		relco 2658	. . 3 |- Rel ((/) o. A)
13		dfrel2 2660	. . 3 |- (Rel ((/) o. A) <-> `'`'((/) o. A) = ((/) o. A))
14	12, 13	mpbi 164	. 2 |- `'`'((/) o. A) = ((/) o. A)
15	8, 11, 14	3eqtr3r 1125	1 |- ((/) o. A) = (/)

Syntax hints:   = wceq 1091  (/)c0 1707  `'ccnv 2409   o. ccom 2414  Rel wrel 2415

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427

metamath.org