Theorem co02 6263

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 6111	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5786	. 2 ⊢ Rel ∅
3		br0 5164	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 492	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1827	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3467	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5858	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 326	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4299	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 378	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5777	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∅c0 4294  ⟨cop 4600   class class class wbr 5113   ∘ ccom 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-co 5671

This theorem is referenced by:  co01  6264  dfpo2  6298  relexpsucld  15070  gsumwmhm  18903  frmdgsum  18920  frmdup1  18922  efginvrel2  19796  0frgp  19848  evl1fval  22456  utop2nei  24375  tngds  24773  tocycf  33377  tocyc01  33378  1arithidom  33771  mrsub0  35906  cononrel1  44211