Theorem co02 6217

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 6065	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5760	. 2 ⊢ Rel ∅
3		br0 5159	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 488	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1802	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3450	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3450	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5832	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 322	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4295	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 375	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5751	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∅c0 4287  ⟨cop 4597   class class class wbr 5110   ∘ ccom 5642

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-co 5647

This theorem is referenced by:  co01  6218  dfpo2  6253  relexpsucld  14931  gsumwmhm  18669  frmdgsum  18686  frmdup1  18688  efginvrel2  19523  0frgp  19575  evl1fval  21731  utop2nei  23639  tngds  24048  tngdsOLD  24049  tocycf  32036  tocyc01  32037  mrsub0  34197  cononrel1  41988