Theorem co01 5216

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 5105	. . . 4 |- `' (/) = (/)
2		cnvco 4881	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4856	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 5215	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2232	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2231	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4871	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4818	. . 3 |- Rel (/)
9		dfrel2 5152	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 145	. 2 |- `' `' (/) = (/)
11		relco 5200	. . 3 |- Rel ( (/) o. A )
12		dfrel2 5152	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 145	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2237	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1373   (/)c0 3468   `'ccnv 4692   o. ccom 4697   Rel wrel 4698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702

This theorem is referenced by: (None)