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Theorem elco 4590
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 4437 . . 3 (𝑅𝑆) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)}
21eleq2i 2154 . 2 (𝐴 ∈ (𝑅𝑆) ↔ 𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)})
3 elopab 4076 . . 3 (𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)} ↔ ∃𝑥𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)))
4 19.42v 1834 . . . . . . 7 (∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ (𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)))
54bicomi 130 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
65exbii 1541 . . . . 5 (∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑧𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
7 excom 1599 . . . . 5 (∃𝑧𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
86, 7bitri 182 . . . 4 (∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
98exbii 1541 . . 3 (∃𝑥𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
103, 9bitri 182 . 2 (𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)} ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
112, 10bitri 182 1 (𝐴 ∈ (𝑅𝑆) ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  cop 3444   class class class wbr 3837  {copab 3890  ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-co 4437
This theorem is referenced by: (None)
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