Theorem co01 4412

Description: Composition with the empty set.

Assertion

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

Proof of Theorem co01

Step	Hyp	Ref	Expression
1		cnv0 4319	. . . . . 6 |- `'(/) = (/)
2	1	coeq2i 4126	. . . . 5 |- (`'A o. `'(/)) = (`'A o. (/))
3		co02 4411	. . . . 5 |- (`'A o. (/)) = (/)
4	2, 3	eqtr2i 1909	. . . 4 |- (/) = (`'A o. `'(/))
5		cnvco 4145	. . . 4 |- `'((/) o. A) = (`'A o. `'(/))
6	4, 1, 5	3eqtr4i 1921	. . 3 |- `'(/) = `'((/) o. A)
7	6	cnveqi 4136	. 2 |- `'`'(/) = `'`'((/) o. A)
8		rel0 4102	. . 3 |- Rel (/)
9		dfrel2 4358	. . 3 |- (Rel (/) <-> `'`'(/) = (/))
10	8, 9	mpbi 206	. 2 |- `'`'(/) = (/)
11		relco 4392	. . 3 |- Rel ((/) o. A)
12		dfrel2 4358	. . 3 |- (Rel ((/) o. A) <-> `'`'((/) o. A) = ((/) o. A))
13	11, 12	mpbi 206	. 2 |- `'`'((/) o. A) = ((/) o. A)
14	7, 10, 13	3eqtr3ri 1920	1 |- ((/) o. A) = (/)

Syntax hints:   = wceq 1298  (/)c0 2875  `'ccnv 3985   o. ccom 3990  Rel wrel 3991

This theorem is referenced by:  empos 14583

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003

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