Theorem co01 5145

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 5034	. . . 4 |- `' (/) = (/)
2		cnvco 4814	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4789	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 5144	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2202	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2201	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4804	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4753	. . 3 |- Rel (/)
9		dfrel2 5081	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 145	. 2 |- `' `' (/) = (/)
11		relco 5129	. . 3 |- Rel ( (/) o. A )
12		dfrel2 5081	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 145	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2207	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1353   (/)c0 3424   `'ccnv 4627   o. ccom 4632   Rel wrel 4633

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637

This theorem is referenced by: (None)