Theorem co02 6164

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 6148	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5709	. 2 ⊢ Rel ∅
3		br0 5123	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 488	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1803	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3436	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3436	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5780	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 323	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4264	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 376	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5700	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∅c0 4256  ⟨cop 4567   class class class wbr 5074   ∘ ccom 5593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-co 5598

This theorem is referenced by:  co01  6165  dfpo2  6199  relexpsucld  14745  gsumwmhm  18484  frmdgsum  18501  frmdup1  18503  efginvrel2  19333  0frgp  19385  evl1fval  21494  utop2nei  23402  tngds  23811  tngdsOLD  23812  tocycf  31384  tocyc01  31385  mrsub0  33478  cononrel1  41202