Theorem co01 5297

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 5186	. . . 4 |- `' (/) = (/)
2		cnvco 4960	. . . . 5 |- `' ( (/) o. A ) = ( `' A o. `' (/) )
3	1	coeq2i 4935	. . . . 5 |- ( `' A o. `' (/) ) = ( `' A o. (/) )
4		co02 5296	. . . . 5 |- ( `' A o. (/) ) = (/)
5	2, 3, 4	3eqtri 2263	. . . 4 |- `' ( (/) o. A ) = (/)
6	1, 5	eqtr4i 2262	. . 3 |- `' (/) = `' ( (/) o. A )
7	6	cnveqi 4950	. 2 |- `' `' (/) = `' `' ( (/) o. A )
8		rel0 4897	. . 3 |- Rel (/)
9		dfrel2 5233	. . 3 |- ( Rel (/) <-> `' `' (/) = (/) )
10	8, 9	mpbi 145	. 2 |- `' `' (/) = (/)
11		relco 5281	. . 3 |- Rel ( (/) o. A )
12		dfrel2 5233	. . 3 |- ( Rel ( (/) o. A ) <-> `' `' ( (/) o. A ) = ( (/) o. A ) )
13	11, 12	mpbi 145	. 2 |- `' `' ( (/) o. A ) = ( (/) o. A )
14	7, 10, 13	3eqtr3ri 2268	1 |- ( (/) o. A ) = (/)

Syntax hints:   = wceq 1402   (/)c0 3520   `'ccnv 4768   o. ccom 4773   Rel wrel 4774

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778

This theorem is referenced by: (None)