Theorem co01 4835

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 4727	. . . 4 |- `' (/) = (/)
2		cnvco 4520	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4496	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 4834	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2064	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2063	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4510	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4462	. . 3 |- Rel (/)
9		dfrel2 4771	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 133	. 2 |- `' `' (/) = (/)
11		relco 4819	. . 3 |- Rel ( (/) o. A )
12		dfrel2 4771	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 133	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2069	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1243   (/)c0 3224   `'ccnv 4344   o. ccom 4349   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354

This theorem is referenced by: (None)