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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 6088 | . 2 ⊢ Rel (𝐴 ∘ I ) | |
2 | vex 3402 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3402 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opelco 5725 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦)) |
5 | vex 3402 | . . . . . . . . . 10 ⊢ 𝑧 ∈ V | |
6 | 5 | ideq 5706 | . . . . . . . . 9 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧) |
7 | equcom 2028 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥) | |
8 | 6, 7 | bitri 278 | . . . . . . . 8 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥) |
9 | 8 | anbi1i 627 | . . . . . . 7 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦)) |
10 | 9 | exbii 1855 | . . . . . 6 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦)) |
11 | breq1 5042 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
12 | 11 | equsexvw 2014 | . . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦) |
13 | 10, 12 | bitri 278 | . . . . 5 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦) |
14 | 4, 13 | bitri 278 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦) |
15 | df-br 5040 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
16 | 14, 15 | bitri 278 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ I ) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
17 | 16 | eqrelriv 5644 | . 2 ⊢ ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴) |
18 | 1, 17 | mpan 690 | 1 ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 〈cop 4533 class class class wbr 5039 I cid 5439 ∘ ccom 5540 Rel wrel 5541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-co 5545 |
This theorem is referenced by: coi2 6107 coires1 6108 fcoi1 6571 mvdco 18791 cocnv 35569 |
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