Theorem co02 6112

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 6096	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5671	. 2 ⊢ Rel ∅
3		br0 5114	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 490	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1797	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3497	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3497	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5741	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 325	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4295	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 378	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5662	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∅c0 4290  ⟨cop 4572   class class class wbr 5065   ∘ ccom 5558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-co 5563

This theorem is referenced by:  co01  6113  gsumwmhm  18009  frmdgsum  18026  frmdup1  18028  efginvrel2  18852  0frgp  18904  evl1fval  20490  utop2nei  22858  tngds  23256  tocycf  30759  tocyc01  30760  mrsub0  32763  dfpo2  32991  cononrel1  39952