Theorem co02 6248

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 6097	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5771	. 2 ⊢ Rel ∅
3		br0 5149	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 491	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1820	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3458	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5843	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 325	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4290	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 377	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5762	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ∅c0 4285  ⟨cop 4588   class class class wbr 5100   ∘ ccom 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-co 5656

This theorem is referenced by:  co01  6249  dfpo2  6283  relexpsucld  15047  gsumwmhm  18879  frmdgsum  18896  frmdup1  18898  efginvrel2  19767  0frgp  19819  evl1fval  22391  utop2nei  24310  tngds  24708  tocycf  33297  tocyc01  33298  1arithidom  33733  mrsub0  35866  cononrel1  44170