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Theorem coi1 6260
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 6106 . 2 Rel (𝐴 ∘ I )
2 vex 3474 . . . . . 6 𝑥 ∈ V
3 vex 3474 . . . . . 6 𝑦 ∈ V
42, 3opelco 5868 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦))
5 vex 3474 . . . . . . . . . 10 𝑧 ∈ V
65ideq 5849 . . . . . . . . 9 (𝑥 I 𝑧𝑥 = 𝑧)
7 equcom 2014 . . . . . . . . 9 (𝑥 = 𝑧𝑧 = 𝑥)
86, 7bitri 275 . . . . . . . 8 (𝑥 I 𝑧𝑧 = 𝑥)
98anbi1i 623 . . . . . . 7 ((𝑥 I 𝑧𝑧𝐴𝑦) ↔ (𝑧 = 𝑥𝑧𝐴𝑦))
109exbii 1843 . . . . . 6 (∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥𝑧𝐴𝑦))
11 breq1 5145 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝐴𝑦𝑥𝐴𝑦))
1211equsexvw 2001 . . . . . 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 5143 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1614, 15bitri 275 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1716eqrelriv 5785 . 2 ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴)
181, 17mpan 689 1 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wex 1774  wcel 2099  cop 4630   class class class wbr 5142   I cid 5569  ccom 5676  Rel wrel 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-co 5681
This theorem is referenced by:  coi2  6261  coires1  6262  fcoi1  6765  mvdco  19393  cocnv  37192
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