Theorem co02 6080

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 6064	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 5636	. 2 ⊢ Rel ∅
3		br0 5079	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
4	3	intnanr 491	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
5	4	nex 1802	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
6		vex 3444	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3444	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	opelco 5706	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
9	5, 8	mtbir 326	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10		noel 4247	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
11	9, 10	2false 379	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
12	1, 2, 11	eqrelriiv 5627	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∅c0 4243  ⟨cop 4531   class class class wbr 5030   ∘ ccom 5523

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-co 5528

This theorem is referenced by:  co01  6081  relexpsucld  14385  gsumwmhm  18002  frmdgsum  18019  frmdup1  18021  efginvrel2  18845  0frgp  18897  evl1fval  20952  utop2nei  22856  tngds  23254  tocycf  30809  tocyc01  30810  mrsub0  32876  dfpo2  33104  cononrel1  40294