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Mirrors > Home > ILE Home > Th. List > elco | GIF version |
Description: Elements of a composed relation. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
elco | ⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-co 4647 | . . 3 ⊢ (𝑅 ∘ 𝑆) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} | |
2 | 1 | eleq2i 2254 | . 2 ⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ 𝐴 ∈ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)}) |
3 | elopab 4270 | . . 3 ⊢ (𝐴 ∈ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} ↔ ∃𝑥∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) | |
4 | 19.42v 1916 | . . . . . . 7 ⊢ (∃𝑦(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ (𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) | |
5 | 4 | bicomi 132 | . . . . . 6 ⊢ ((𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
6 | 5 | exbii 1615 | . . . . 5 ⊢ (∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑧∃𝑦(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
7 | excom 1674 | . . . . 5 ⊢ (∃𝑧∃𝑦(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) | |
8 | 6, 7 | bitri 184 | . . . 4 ⊢ (∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
9 | 8 | exbii 1615 | . . 3 ⊢ (∃𝑥∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
10 | 3, 9 | bitri 184 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
11 | 2, 10 | bitri 184 | 1 ⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1363 ∃wex 1502 ∈ wcel 2158 〈cop 3607 class class class wbr 4015 {copab 4075 ∘ ccom 4642 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-opab 4077 df-co 4647 |
This theorem is referenced by: (None) |
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