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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 5878 | . 2 ⊢ Rel (𝐴 ∘ I ) | |
2 | vex 3417 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3417 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opelco 5530 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦)) |
5 | vex 3417 | . . . . . . . . . 10 ⊢ 𝑧 ∈ V | |
6 | 5 | ideq 5511 | . . . . . . . . 9 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧) |
7 | equcom 2122 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥) | |
8 | 6, 7 | bitri 267 | . . . . . . . 8 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥) |
9 | 8 | anbi1i 617 | . . . . . . 7 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦)) |
10 | 9 | exbii 1947 | . . . . . 6 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦)) |
11 | breq1 4878 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
12 | 11 | equsexvw 2109 | . . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦) |
13 | 10, 12 | bitri 267 | . . . . 5 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦) |
14 | 4, 13 | bitri 267 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦) |
15 | df-br 4876 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
16 | 14, 15 | bitri 267 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ I ) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
17 | 16 | eqrelriv 5451 | . 2 ⊢ ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴) |
18 | 1, 17 | mpan 681 | 1 ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∃wex 1878 ∈ wcel 2164 〈cop 4405 class class class wbr 4875 I cid 5251 ∘ ccom 5350 Rel wrel 5351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-co 5355 |
This theorem is referenced by: coi2 5897 coires1 5898 fcoi1 6319 mvdco 18222 cocnv 34062 |
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