Theorem co01 3509

Description: Composition with the empty set.

Assertion

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

Proof of Theorem co01

Step	Hyp	Ref	Expression
1		cnv0 3446	. . . . . 6 |- `'(/) = (/)
2	1	coeq2i 3284	. . . . 5 |- (`'A o. `'(/)) = (`'A o. (/))
3		co02 3508	. . . . 5 |- (`'A o. (/)) = (/)
4	2, 3	eqtr2 1496	. . . 4 |- (/) = (`'A o. `'(/))
5		cnvco 3300	. . . 4 |- `'((/) o. A) = (`'A o. `'(/))
6	4, 1, 5	3eqtr4 1505	. . 3 |- `'(/) = `'((/) o. A)
7		cnveq 3292	. . 3 |- (`'(/) = `'((/) o. A) -> `'`'(/) = `'`'((/) o. A))
8	6, 7	ax-mp 7	. 2 |- `'`'(/) = `'`'((/) o. A)
9		rel0 3272	. . 3 |- Rel (/)
10		dfrel2 3485	. . 3 |- (Rel (/) <-> `'`'(/) = (/))
11	9, 10	mpbi 189	. 2 |- `'`'(/) = (/)
12		relco 3484	. . 3 |- Rel ((/) o. A)
13		dfrel2 3485	. . 3 |- (Rel ((/) o. A) <-> `'`'((/) o. A) = ((/) o. A))
14	12, 13	mpbi 189	. 2 |- `'`'((/) o. A) = ((/) o. A)
15	8, 11, 14	3eqtr3r 1504	1 |- ((/) o. A) = (/)

Syntax hints:   = wceq 956  (/)c0 2280  `'ccnv 3169   o. ccom 3174  Rel wrel 3175

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187

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