Theorem co01 5277

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 5166	. . . 4 |- `' (/) = (/)
2		cnvco 4940	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4915	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 5276	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2257	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2256	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4930	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4877	. . 3 |- Rel (/)
9		dfrel2 5213	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 145	. 2 |- `' `' (/) = (/)
11		relco 5261	. . 3 |- Rel ( (/) o. A )
12		dfrel2 5213	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 145	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2262	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1398   (/)c0 3508   `'ccnv 4748   o. ccom 4753   Rel wrel 4754

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758

This theorem is referenced by: (None)