Theorem co02 6282

Description: Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)

Assertion

Ref	Expression
co02	⊢ (𝐴 ∘ ∅) = ∅

Proof of Theorem co02

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 6129	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5812	. 2 ⊢ Rel ∅
3		br0 5197	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 487	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1797	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3482	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3482	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5885	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 323	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4344	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 375	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5803	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∅c0 4339  ⟨cop 4637   class class class wbr 5148   ∘ ccom 5693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-co 5698

This theorem is referenced by:  co01  6283  dfpo2  6318  relexpsucld  15070  gsumwmhm  18871  frmdgsum  18888  frmdup1  18890  efginvrel2  19760  0frgp  19812  evl1fval  22348  utop2nei  24275  tngds  24684  tngdsOLD  24685  tocycf  33120  tocyc01  33121  1arithidom  33545  mrsub0  35501  cononrel1  43584