Theorem co01 5197

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 5086	. . . 4 |- `' (/) = (/)
2		cnvco 4863	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4838	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 5196	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2230	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2229	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4853	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4800	. . 3 |- Rel (/)
9		dfrel2 5133	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 145	. 2 |- `' `' (/) = (/)
11		relco 5181	. . 3 |- Rel ( (/) o. A )
12		dfrel2 5133	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 145	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2235	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1373   (/)c0 3460   `'ccnv 4674   o. ccom 4679   Rel wrel 4680

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684

This theorem is referenced by: (None)