Theorem co01 5155

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 5044	. . . 4 |- `' (/) = (/)
2		cnvco 4824	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4799	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 5154	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2212	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2211	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4814	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4763	. . 3 |- Rel (/)
9		dfrel2 5091	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 145	. 2 |- `' `' (/) = (/)
11		relco 5139	. . 3 |- Rel ( (/) o. A )
12		dfrel2 5091	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 145	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2217	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1363   (/)c0 3434   `'ccnv 4637   o. ccom 4642   Rel wrel 4643

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647

This theorem is referenced by: (None)