Theorem co02 6221

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 6068	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5753	. 2 ⊢ Rel ∅
3		br0 5151	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 487	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1800	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3448	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3448	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5825	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 323	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4297	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 375	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5744	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∅c0 4292  ⟨cop 4591   class class class wbr 5102   ∘ ccom 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-co 5640

This theorem is referenced by:  co01  6222  dfpo2  6257  relexpsucld  14976  gsumwmhm  18748  frmdgsum  18765  frmdup1  18767  efginvrel2  19633  0frgp  19685  evl1fval  22191  utop2nei  24114  tngds  24512  tocycf  33047  tocyc01  33048  1arithidom  33481  mrsub0  35476  cononrel1  43556