Theorem co02 6283

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 6122	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5785	. 2 ⊢ Rel ∅
3		br0 5160	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 493	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1833	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3458	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5861	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 326	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4286	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 378	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5775	1 ⊢ (𝐴 ∘ ∅) = ∅

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∅c0 4281  ⟨cop 4840   class class class wbr 5108   ∘ ccom 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-co 5667

This theorem is used by:  co01  6284  dfpo2  6325  relexpsucld  15230  gsumwmhm  19091  frmdgsum  19108  frmdup1  19110  efginvrel2  19992  0frgp  20044  evl1fval  22701  utop2nei  24624  tngds  25022  tocycf  33777  tocyc01  33778  1arithidom  34173  mrsub0  36489  coineq0  38208  cononrel1  44855