Theorem co01 5061

Description: Composition with the empty set. (Contributed by NM, 24-Apr-2004.)

Assertion

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

Proof of Theorem co01

Step	Hyp	Ref	Expression
1		cnv0 4950	. . . 4 |- `' (/) = (/)
2		cnvco 4732	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4707	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 5060	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2165	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2164	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4722	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4672	. . 3 |- Rel (/)
9		dfrel2 4997	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 144	. 2 |- `' `' (/) = (/)
11		relco 5045	. . 3 |- Rel ( (/) o. A )
12		dfrel2 4997	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 144	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2170	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1332   (/)c0 3368   `'ccnv 4546   o. ccom 4551   Rel wrel 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556

This theorem is referenced by: (None)