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Theorem elco 4749
 Description: Elements of a composed relation. (Contributed by BJ, 10-Jul-2022.)
Assertion
Ref Expression
elco (𝐴 ∈ (𝑅𝑆) ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
Distinct variable groups:   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧

Proof of Theorem elco
StepHypRef Expression
1 df-co 4592 . . 3 (𝑅𝑆) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)}
21eleq2i 2224 . 2 (𝐴 ∈ (𝑅𝑆) ↔ 𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)})
3 elopab 4217 . . 3 (𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)} ↔ ∃𝑥𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)))
4 19.42v 1886 . . . . . . 7 (∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ (𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)))
54bicomi 131 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
65exbii 1585 . . . . 5 (∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑧𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
7 excom 1644 . . . . 5 (∃𝑧𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
86, 7bitri 183 . . . 4 (∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
98exbii 1585 . . 3 (∃𝑥𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
103, 9bitri 183 . 2 (𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)} ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
112, 10bitri 183 1 (𝐴 ∈ (𝑅𝑆) ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ⟨cop 3563   class class class wbr 3965  {copab 4024   ∘ ccom 4587 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-co 4592 This theorem is referenced by: (None)
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