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Mirrors > Home > ILE Home > Th. List > coi1 | GIF version |
Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Ref | Expression |
---|---|
coi1 | ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 5128 | . 2 ⊢ Rel (𝐴 ∘ I ) | |
2 | vex 2741 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 2741 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opelco 4800 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦)) |
5 | vex 2741 | . . . . . . . . . 10 ⊢ 𝑧 ∈ V | |
6 | 5 | ideq 4780 | . . . . . . . . 9 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧) |
7 | equcom 1706 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥) | |
8 | 6, 7 | bitri 184 | . . . . . . . 8 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥) |
9 | 8 | anbi1i 458 | . . . . . . 7 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦)) |
10 | 9 | exbii 1605 | . . . . . 6 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦)) |
11 | breq1 4007 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
12 | 2, 11 | ceqsexv 2777 | . . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦) |
13 | 10, 12 | bitri 184 | . . . . 5 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦) |
14 | 4, 13 | bitri 184 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦) |
15 | df-br 4005 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
16 | 14, 15 | bitri 184 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) |
17 | 16 | eqrelriv 4720 | . 2 ⊢ ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴) |
18 | 1, 17 | mpan 424 | 1 ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ⟨cop 3596 class class class wbr 4004 I cid 4289 ∘ ccom 4631 Rel wrel 4632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-br 4005 df-opab 4066 df-id 4294 df-xp 4633 df-rel 4634 df-co 4636 |
This theorem is referenced by: coi2 5146 coires1 5147 relcoi1 5161 fcoi1 5397 |
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