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Theorem coi1 4863
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 4846 . 2 Rel (𝐴 ∘ I )
2 vex 2577 . . . . . 6 𝑥 ∈ V
3 vex 2577 . . . . . 6 𝑦 ∈ V
42, 3opelco 4534 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦))
5 vex 2577 . . . . . . . . . 10 𝑧 ∈ V
65ideq 4515 . . . . . . . . 9 (𝑥 I 𝑧𝑥 = 𝑧)
7 equcom 1609 . . . . . . . . 9 (𝑥 = 𝑧𝑧 = 𝑥)
86, 7bitri 177 . . . . . . . 8 (𝑥 I 𝑧𝑧 = 𝑥)
98anbi1i 439 . . . . . . 7 ((𝑥 I 𝑧𝑧𝐴𝑦) ↔ (𝑧 = 𝑥𝑧𝐴𝑦))
109exbii 1512 . . . . . 6 (∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥𝑧𝐴𝑦))
11 breq1 3794 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝐴𝑦𝑥𝐴𝑦))
122, 11ceqsexv 2610 . . . . . 6 (∃𝑧(𝑧 = 𝑥𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
1310, 12bitri 177 . . . . 5 (∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
144, 13bitri 177 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦)
15 df-br 3792 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1614, 15bitri 177 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1716eqrelriv 4460 . 2 ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴)
181, 17mpan 408 1 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  cop 3405   class class class wbr 3791   I cid 4052  ccom 4376  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-co 4381
This theorem is referenced by:  coi2  4864  coires1  4865  relcoi1  4876  fcoi1  5097
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