Theorem co01 6257

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 6137	. . . 4 ⊢ ◡∅ = ∅
2		cnvco 5883	. . . . 5 ⊢ ◡(∅ ∘ 𝐴) = (◡𝐴 ∘ ◡∅)
3	1	coeq2i 5858	. . . . 5 ⊢ (◡𝐴 ∘ ◡∅) = (◡𝐴 ∘ ∅)
4		co02 6256	. . . . 5 ⊢ (◡𝐴 ∘ ∅) = ∅
5	2, 3, 4	3eqtri 2764	. . . 4 ⊢ ◡(∅ ∘ 𝐴) = ∅
6	1, 5	eqtr4i 2763	. . 3 ⊢ ◡∅ = ◡(∅ ∘ 𝐴)
7	6	cnveqi 5872	. 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ 𝐴)
8		rel0 5797	. . 3 ⊢ Rel ∅
9		dfrel2 6185	. . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅)
10	8, 9	mpbi 229	. 2 ⊢ ◡◡∅ = ∅
11		relco 6104	. . 3 ⊢ Rel (∅ ∘ 𝐴)
12		dfrel2 6185	. . 3 ⊢ (Rel (∅ ∘ 𝐴) ↔ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴))
13	11, 12	mpbi 229	. 2 ⊢ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴)
14	7, 10, 13	3eqtr3ri 2769	1 ⊢ (∅ ∘ 𝐴) = ∅

Syntax hints:   = wceq 1541  ∅c0 4321  ^◡ccnv 5674   ∘ ccom 5679  Rel wrel 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684

This theorem is referenced by:  xpcoid  6286  0trrel  14924  relexpsucrd  14976  relexpaddd  14997  gsumval3  19769  utop2nei  23746  cononrel2  42331