Theorem co02 6260

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 6108	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5800	. 2 ⊢ Rel ∅
3		br0 5198	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 489	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1803	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3479	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3479	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5872	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 323	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4331	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 376	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5791	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∅c0 4323  ⟨cop 4635   class class class wbr 5149   ∘ ccom 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-co 5686

This theorem is referenced by:  co01  6261  dfpo2  6296  relexpsucld  14981  gsumwmhm  18726  frmdgsum  18743  frmdup1  18745  efginvrel2  19595  0frgp  19647  evl1fval  21847  utop2nei  23755  tngds  24164  tngdsOLD  24165  tocycf  32276  tocyc01  32277  mrsub0  34507  cononrel1  42345