Theorem co01 6282

Description: Composition with the empty set. (Contributed by NM, 24-Apr-2004.)

Assertion

Ref	Expression
co01	⊢ (∅ ∘ 𝐴) = ∅

Proof of Theorem co01

Step	Hyp	Ref	Expression
1		cnv0 6162	. . . 4 ⊢ ◡∅ = ∅
2		cnvco 5898	. . . . 5 ⊢ ◡(∅ ∘ 𝐴) = (◡𝐴 ∘ ◡∅)
3	1	coeq2i 5873	. . . . 5 ⊢ (◡𝐴 ∘ ◡∅) = (◡𝐴 ∘ ∅)
4		co02 6281	. . . . 5 ⊢ (◡𝐴 ∘ ∅) = ∅
5	2, 3, 4	3eqtri 2766	. . . 4 ⊢ ◡(∅ ∘ 𝐴) = ∅
6	1, 5	eqtr4i 2765	. . 3 ⊢ ◡∅ = ◡(∅ ∘ 𝐴)
7	6	cnveqi 5887	. 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ 𝐴)
8		rel0 5811	. . 3 ⊢ Rel ∅
9		dfrel2 6210	. . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅)
10	8, 9	mpbi 230	. 2 ⊢ ◡◡∅ = ∅
11		relco 6128	. . 3 ⊢ Rel (∅ ∘ 𝐴)
12		dfrel2 6210	. . 3 ⊢ (Rel (∅ ∘ 𝐴) ↔ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴))
13	11, 12	mpbi 230	. 2 ⊢ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴)
14	7, 10, 13	3eqtr3ri 2771	1 ⊢ (∅ ∘ 𝐴) = ∅

Syntax hints:   = wceq 1536  ∅c0 4338  ^◡ccnv 5687   ∘ ccom 5692  Rel wrel 5693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697

This theorem is referenced by:  xpcoid  6311  0trrel  15016  relexpsucrd  15068  relexpaddd  15089  gsumval3  19939  utop2nei  24274  cononrel2  43584