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Theorem elco 4665
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 4508 . . 3 (𝑅𝑆) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)}
21eleq2i 2181 . 2 (𝐴 ∈ (𝑅𝑆) ↔ 𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)})
3 elopab 4140 . . 3 (𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)} ↔ ∃𝑥𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)))
4 19.42v 1860 . . . . . . 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 1567 . . . . 5 (∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦𝑦𝑅𝑧)) ↔ ∃𝑧𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
7 excom 1625 . . . . 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 1567 . . 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 1314  wex 1451  wcel 1463  cop 3496   class class class wbr 3895  {copab 3948  ccom 4503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-opab 3950  df-co 4508
This theorem is referenced by: (None)
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