Theorem co02 6291

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 6138	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5823	. 2 ⊢ Rel ∅
3		br0 5215	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 487	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1798	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3492	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3492	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5896	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 323	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4360	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 375	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5814	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∅c0 4352  ⟨cop 4654   class class class wbr 5166   ∘ ccom 5704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-co 5709

This theorem is referenced by:  co01  6292  dfpo2  6327  relexpsucld  15083  gsumwmhm  18880  frmdgsum  18897  frmdup1  18899  efginvrel2  19769  0frgp  19821  evl1fval  22353  utop2nei  24280  tngds  24689  tngdsOLD  24690  tocycf  33110  tocyc01  33111  1arithidom  33530  mrsub0  35484  cononrel1  43556