Theorem elco 4805

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 4647	. . 3 ⊢ (𝑅 ∘ 𝑆) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)}
2	1	eleq2i 2254	. 2 ⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ 𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)})
3		elopab 4270	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} ↔ ∃𝑥∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
4		19.42v 1916	. . . . . . 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 1615	. . . . 5 ⊢ (∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑧∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
7		excom 1674	. . . . 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 1615	. . 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 1363  ∃wex 1502   ∈ wcel 2158  ⟨cop 3607   class class class wbr 4015  {copab 4075   ∘ ccom 4642

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-opab 4077  df-co 4647

This theorem is referenced by: (None)