Theorem co02 5296

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 5281	. 2 ⊢ Rel (𝐴 ∘ ∅)
2		rel0 4897	. 2 ⊢ Rel ∅
3		noel 3525	. . . . . . 7 ⊢ ¬ ⟨𝑥, 𝑧⟩ ∈ ∅
4		df-br 4126	. . . . . . 7 ⊢ (𝑥∅𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ ∅)
5	3, 4	mtbir 682	. . . . . 6 ⊢ ¬ 𝑥∅𝑧
6	5	intnanr 942	. . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
7	6	nex 1553	. . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)
8		vex 2824	. . . . 5 ⊢ 𝑥 ∈ V
9		vex 2824	. . . . 5 ⊢ 𝑦 ∈ V
10	8, 9	opelco 4947	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦))
11	7, 10	mtbir 682	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
12		noel 3525	. . 3 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
13	11, 12	2false 713	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
14	1, 2, 13	eqrelriiv 4864	1 ⊢ (𝐴 ∘ ∅) = ∅

Syntax hints:   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∅c0 3520  ⟨cop 3708   class class class wbr 4125   ∘ ccom 4773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-co 4778

This theorem is referenced by:  co01  5297  gzsumwmhm  13780  gsumvalfi  14129