Theorem co02 6225

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 6073	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5755	. 2 ⊢ Rel ∅
3		br0 5134	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 487	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1802	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3433	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3433	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5826	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 323	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4278	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 375	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5746	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∅c0 4273  ⟨cop 4573   class class class wbr 5085   ∘ ccom 5635

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-co 5640

This theorem is referenced by:  co01  6226  dfpo2  6260  relexpsucld  14996  gsumwmhm  18813  frmdgsum  18830  frmdup1  18832  efginvrel2  19702  0frgp  19754  evl1fval  22293  utop2nei  24215  tngds  24613  tocycf  33178  tocyc01  33179  1arithidom  33597  mrsub0  35698  cononrel1  44021