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Theorem coi1 6252
Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
Assertion
Ref Expression
coi1 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)

Proof of Theorem coi1
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 6098 . 2 Rel (𝐴 ∘ I )
2 vex 3470 . . . . . 6 𝑥 ∈ V
3 vex 3470 . . . . . 6 𝑦 ∈ V
42, 3opelco 5862 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦))
5 vex 3470 . . . . . . . . . 10 𝑧 ∈ V
65ideq 5843 . . . . . . . . 9 (𝑥 I 𝑧𝑥 = 𝑧)
7 equcom 2013 . . . . . . . . 9 (𝑥 = 𝑧𝑧 = 𝑥)
86, 7bitri 275 . . . . . . . 8 (𝑥 I 𝑧𝑧 = 𝑥)
98anbi1i 623 . . . . . . 7 ((𝑥 I 𝑧𝑧𝐴𝑦) ↔ (𝑧 = 𝑥𝑧𝐴𝑦))
109exbii 1842 . . . . . 6 (∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥𝑧𝐴𝑦))
11 breq1 5142 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝐴𝑦𝑥𝐴𝑦))
1211equsexvw 2000 . . . . . 6 (∃𝑧(𝑧 = 𝑥𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
1310, 12bitri 275 . . . . 5 (∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
144, 13bitri 275 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦)
15 df-br 5140 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1614, 15bitri 275 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1716eqrelriv 5780 . 2 ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴)
181, 17mpan 687 1 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wex 1773  wcel 2098  cop 4627   class class class wbr 5139   I cid 5564  ccom 5671  Rel wrel 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-co 5676
This theorem is referenced by:  coi2  6253  coires1  6254  fcoi1  6756  mvdco  19357  cocnv  37087
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