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Theorem coi1 4889
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 4872 . 2 Rel (𝐴 ∘ I )
2 vex 2614 . . . . . 6 𝑥 ∈ V
3 vex 2614 . . . . . 6 𝑦 ∈ V
42, 3opelco 4558 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦))
5 vex 2614 . . . . . . . . . 10 𝑧 ∈ V
65ideq 4539 . . . . . . . . 9 (𝑥 I 𝑧𝑥 = 𝑧)
7 equcom 1635 . . . . . . . . 9 (𝑥 = 𝑧𝑧 = 𝑥)
86, 7bitri 182 . . . . . . . 8 (𝑥 I 𝑧𝑧 = 𝑥)
98anbi1i 446 . . . . . . 7 ((𝑥 I 𝑧𝑧𝐴𝑦) ↔ (𝑧 = 𝑥𝑧𝐴𝑦))
109exbii 1537 . . . . . 6 (∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥𝑧𝐴𝑦))
11 breq1 3809 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝐴𝑦𝑥𝐴𝑦))
122, 11ceqsexv 2648 . . . . . 6 (∃𝑧(𝑧 = 𝑥𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
1310, 12bitri 182 . . . . 5 (∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
144, 13bitri 182 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦)
15 df-br 3807 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1614, 15bitri 182 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1716eqrelriv 4482 . 2 ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴)
181, 17mpan 415 1 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  cop 3420   class class class wbr 3806   I cid 4073  ccom 4398  Rel wrel 4399
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3994
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-id 4078  df-xp 4400  df-rel 4401  df-co 4403
This theorem is referenced by:  coi2  4890  coires1  4891  relcoi1  4902  fcoi1  5124
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