Theorem co01 5118

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 5007	. . . 4 |- `' (/) = (/)
2		cnvco 4789	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4764	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 5117	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2190	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2189	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4779	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4729	. . 3 |- Rel (/)
9		dfrel2 5054	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 144	. 2 |- `' `' (/) = (/)
11		relco 5102	. . 3 |- Rel ( (/) o. A )
12		dfrel2 5054	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 144	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2195	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1343   (/)c0 3409   `'ccnv 4603   o. ccom 4608   Rel wrel 4609

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613

This theorem is referenced by: (None)