Theorem co01 4865

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 4757	. . . 4 |- `' (/) = (/)
2		cnvco 4548	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4524	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 4864	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2106	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2105	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4538	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4490	. . 3 |- Rel (/)
9		dfrel2 4801	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 143	. 2 |- `' `' (/) = (/)
11		relco 4849	. . 3 |- Rel ( (/) o. A )
12		dfrel2 4801	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 143	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2111	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1285   (/)c0 3258   `'ccnv 4370   o. ccom 4375   Rel wrel 4376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380

This theorem is referenced by: (None)