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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 6091 | . 2 ⊢ Rel (𝐴 ∘ I ) | |
2 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opelco 5736 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦)) |
5 | vex 3497 | . . . . . . . . . 10 ⊢ 𝑧 ∈ V | |
6 | 5 | ideq 5717 | . . . . . . . . 9 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧) |
7 | equcom 2021 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥) | |
8 | 6, 7 | bitri 277 | . . . . . . . 8 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥) |
9 | 8 | anbi1i 625 | . . . . . . 7 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦)) |
10 | 9 | exbii 1844 | . . . . . 6 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦)) |
11 | breq1 5061 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
12 | 11 | equsexvw 2007 | . . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦) |
13 | 10, 12 | bitri 277 | . . . . 5 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦) |
14 | 4, 13 | bitri 277 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦) |
15 | df-br 5059 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
16 | 14, 15 | bitri 277 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ I ) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
17 | 16 | eqrelriv 5656 | . 2 ⊢ ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴) |
18 | 1, 17 | mpan 688 | 1 ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 I cid 5453 ∘ ccom 5553 Rel wrel 5554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-co 5558 |
This theorem is referenced by: coi2 6110 coires1 6111 fcoi1 6546 mvdco 18567 cocnv 34994 |
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