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Mirrors > Home > MPE Home > Th. List > coi1 | Structured version Visualization version 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 6098 | . 2 ⊢ Rel (𝐴 ∘ I ) | |
2 | vex 3470 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3470 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opelco 5862 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦)) |
5 | vex 3470 | . . . . . . . . . 10 ⊢ 𝑧 ∈ V | |
6 | 5 | ideq 5843 | . . . . . . . . 9 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧) |
7 | equcom 2013 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥) | |
8 | 6, 7 | bitri 275 | . . . . . . . 8 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥) |
9 | 8 | anbi1i 623 | . . . . . . 7 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦)) |
10 | 9 | exbii 1842 | . . . . . 6 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦)) |
11 | breq1 5142 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
12 | 11 | equsexvw 2000 | . . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦) |
13 | 10, 12 | bitri 275 | . . . . 5 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦) |
14 | 4, 13 | bitri 275 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦) |
15 | df-br 5140 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
16 | 14, 15 | bitri 275 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) |
17 | 16 | eqrelriv 5780 | . 2 ⊢ ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴) |
18 | 1, 17 | mpan 687 | 1 ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ⟨cop 4627 class class class wbr 5139 I cid 5564 ∘ ccom 5671 Rel wrel 5672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-co 5676 |
This theorem is referenced by: coi2 6253 coires1 6254 fcoi1 6756 mvdco 19357 cocnv 37087 |
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