Theorem co02 6113

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 5672	. 2 ⊢ Rel ∅
3		br0 5115	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 490	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1801	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3497	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3497	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5742	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 325	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4296	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 378	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5663	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∅c0 4291  ⟨cop 4573   class class class wbr 5066   ∘ ccom 5559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-co 5564

This theorem is referenced by:  co01  6114  gsumwmhm  18010  frmdgsum  18027  frmdup1  18029  efginvrel2  18853  0frgp  18905  evl1fval  20491  utop2nei  22859  tngds  23257  tocycf  30759  tocyc01  30760  mrsub0  32763  dfpo2  32991  cononrel1  39974