Theorem co01 6162

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 6041	. . . 4 ⊢ ◡∅ = ∅
2		cnvco 5791	. . . . 5 ⊢ ◡(∅ ∘ 𝐴) = (◡𝐴 ∘ ◡∅)
3	1	coeq2i 5766	. . . . 5 ⊢ (◡𝐴 ∘ ◡∅) = (◡𝐴 ∘ ∅)
4		co02 6161	. . . . 5 ⊢ (◡𝐴 ∘ ∅) = ∅
5	2, 3, 4	3eqtri 2771	. . . 4 ⊢ ◡(∅ ∘ 𝐴) = ∅
6	1, 5	eqtr4i 2770	. . 3 ⊢ ◡∅ = ◡(∅ ∘ 𝐴)
7	6	cnveqi 5780	. 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ 𝐴)
8		rel0 5706	. . 3 ⊢ Rel ∅
9		dfrel2 6089	. . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅)
10	8, 9	mpbi 229	. 2 ⊢ ◡◡∅ = ∅
11		relco 6145	. . 3 ⊢ Rel (∅ ∘ 𝐴)
12		dfrel2 6089	. . 3 ⊢ (Rel (∅ ∘ 𝐴) ↔ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴))
13	11, 12	mpbi 229	. 2 ⊢ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴)
14	7, 10, 13	3eqtr3ri 2776	1 ⊢ (∅ ∘ 𝐴) = ∅

Syntax hints:   = wceq 1541  ∅c0 4261  ^◡ccnv 5587   ∘ ccom 5592  Rel wrel 5593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597

This theorem is referenced by:  xpcoid  6190  0trrel  14673  relexpsucrd  14725  relexpaddd  14746  gsumval3  19489  utop2nei  23383  cononrel2  41156