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Mirrors > Home > ILE Home > Th. List > elreldm | GIF version |
Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.) |
Ref | Expression |
---|---|
elreldm | ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 4546 | . . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
2 | ssel 3091 | . . . . 5 ⊢ (𝐴 ⊆ (V × V) → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) | |
3 | 1, 2 | sylbi 120 | . . . 4 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) |
4 | elvv 4601 | . . . 4 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) | |
5 | 3, 4 | syl6ib 160 | . . 3 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉)) |
6 | eleq1 2202 | . . . . . 6 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
7 | vex 2689 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | vex 2689 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opeldm 4742 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
10 | 6, 9 | syl6bi 162 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → 𝑥 ∈ dom 𝐴)) |
11 | inteq 3774 | . . . . . . . 8 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ 𝐵 = ∩ 〈𝑥, 𝑦〉) | |
12 | 11 | inteqd 3776 | . . . . . . 7 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐵 = ∩ ∩ 〈𝑥, 𝑦〉) |
13 | 7, 8 | op1stb 4399 | . . . . . . 7 ⊢ ∩ ∩ 〈𝑥, 𝑦〉 = 𝑥 |
14 | 12, 13 | syl6eq 2188 | . . . . . 6 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐵 = 𝑥) |
15 | 14 | eleq1d 2208 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (∩ ∩ 𝐵 ∈ dom 𝐴 ↔ 𝑥 ∈ dom 𝐴)) |
16 | 10, 15 | sylibrd 168 | . . . 4 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴)) |
17 | 16 | exlimivv 1868 | . . 3 ⊢ (∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴)) |
18 | 5, 17 | syli 37 | . 2 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴)) |
19 | 18 | imp 123 | 1 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 〈cop 3530 ∩ cint 3771 × cxp 4537 dom cdm 4539 Rel wrel 4544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-int 3772 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-dm 4549 |
This theorem is referenced by: 1stdm 6080 fundmen 6700 |
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