Theorem elco 4828

Description: Elements of a composed relation. (Contributed by BJ, 10-Jul-2022.)

Assertion

Ref	Expression
elco	⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))

Distinct variable groups:   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧

Proof of Theorem elco

Step	Hyp	Ref	Expression
1		df-co 4668	. . 3 ⊢ (𝑅 ∘ 𝑆) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)}
2	1	eleq2i 2260	. 2 ⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ 𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)})
3		elopab 4288	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} ↔ ∃𝑥∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
4		19.42v 1918	. . . . . . 7 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ (𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
5	4	bicomi 132	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
6	5	exbii 1616	. . . . 5 ⊢ (∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑧∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
7		excom 1675	. . . . 5 ⊢ (∃𝑧∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
8	6, 7	bitri 184	. . . 4 ⊢ (∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
9	8	exbii 1616	. . 3 ⊢ (∃𝑥∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
10	3, 9	bitri 184	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
11	2, 10	bitri 184	1 ⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ⟨cop 3621   class class class wbr 4029  {copab 4089   ∘ ccom 4663

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091  df-co 4668

This theorem is referenced by: (None)