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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 4508 | . . 3 ⊢ (𝑅 ∘ 𝑆) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} | |
2 | 1 | eleq2i 2181 | . 2 ⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ 𝐴 ∈ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)}) |
3 | elopab 4140 | . . 3 ⊢ (𝐴 ∈ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} ↔ ∃𝑥∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) | |
4 | 19.42v 1860 | . . . . . . 7 ⊢ (∃𝑦(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ (𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) | |
5 | 4 | bicomi 131 | . . . . . 6 ⊢ ((𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
6 | 5 | exbii 1567 | . . . . 5 ⊢ (∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑧∃𝑦(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
7 | excom 1625 | . . . . 5 ⊢ (∃𝑧∃𝑦(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) | |
8 | 6, 7 | bitri 183 | . . . 4 ⊢ (∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
9 | 8 | exbii 1567 | . . 3 ⊢ (∃𝑥∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
10 | 3, 9 | bitri 183 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
11 | 2, 10 | bitri 183 | 1 ⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = 〈𝑥, 𝑧〉 ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1314 ∃wex 1451 ∈ wcel 1463 〈cop 3496 class class class wbr 3895 {copab 3948 ∘ ccom 4503 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-opab 3950 df-co 4508 |
This theorem is referenced by: (None) |
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