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Theorem coi1 5794
 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 5776 . 2 Rel (𝐴 ∘ I )
2 vex 3354 . . . . . 6 𝑥 ∈ V
3 vex 3354 . . . . . 6 𝑦 ∈ V
42, 3opelco 5431 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦))
5 vex 3354 . . . . . . . . . 10 𝑧 ∈ V
65ideq 5412 . . . . . . . . 9 (𝑥 I 𝑧𝑥 = 𝑧)
7 equcom 2103 . . . . . . . . 9 (𝑥 = 𝑧𝑧 = 𝑥)
86, 7bitri 264 . . . . . . . 8 (𝑥 I 𝑧𝑧 = 𝑥)
98anbi1i 610 . . . . . . 7 ((𝑥 I 𝑧𝑧𝐴𝑦) ↔ (𝑧 = 𝑥𝑧𝐴𝑦))
109exbii 1924 . . . . . 6 (∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥𝑧𝐴𝑦))
11 breq1 4790 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝐴𝑦𝑥𝐴𝑦))
1211equsexvw 2090 . . . . . 6 (∃𝑧(𝑧 = 𝑥𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
1310, 12bitri 264 . . . . 5 (∃𝑧(𝑥 I 𝑧𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
144, 13bitri 264 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦)
15 df-br 4788 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1614, 15bitri 264 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1716eqrelriv 5352 . 2 ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴)
181, 17mpan 670 1 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ⟨cop 4323   class class class wbr 4787   I cid 5157   ∘ ccom 5254  Rel wrel 5255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-co 5259 This theorem is referenced by:  coi2  5795  coires1  5796  fcoi1  6219  mvdco  18072  cocnv  33852
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