Theorem co01 4855

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 4747	. . . 4 |- `' (/) = (/)
2		cnvco 4538	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4514	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 4854	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2105	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2104	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4528	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4480	. . 3 |- Rel (/)
9		dfrel2 4791	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 143	. 2 |- `' `' (/) = (/)
11		relco 4839	. . 3 |- Rel ( (/) o. A )
12		dfrel2 4791	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 143	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2110	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1284   (/)c0 3251   `'ccnv 4362   o. ccom 4367   Rel wrel 4368

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372

This theorem is referenced by: (None)