Theorem co02 6094

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 6078	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5645	. 2 ⊢ Rel ∅
3		br0 5084	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 491	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1802	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3413	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3413	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5716	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 326	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4232	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 379	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5636	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∅c0 4227  ⟨cop 4531   class class class wbr 5035   ∘ ccom 5531

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-co 5536

This theorem is referenced by:  co01  6095  relexpsucld  14446  gsumwmhm  18081  frmdgsum  18098  frmdup1  18100  efginvrel2  18925  0frgp  18977  evl1fval  21052  utop2nei  22956  tngds  23355  tocycf  30914  tocyc01  30915  mrsub0  32998  dfpo2  33242  cononrel1  40695