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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 4370 | . . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
2 | ssel 2993 | . . . . 5 ⊢ (𝐴 ⊆ (V × V) → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) | |
3 | 1, 2 | sylbi 119 | . . . 4 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) |
4 | elvv 4420 | . . . 4 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) | |
5 | 3, 4 | syl6ib 159 | . . 3 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉)) |
6 | eleq1 2141 | . . . . . 6 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
7 | vex 2604 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | vex 2604 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opeldm 4556 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
10 | 6, 9 | syl6bi 161 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → 𝑥 ∈ dom 𝐴)) |
11 | inteq 3639 | . . . . . . . 8 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ 𝐵 = ∩ 〈𝑥, 𝑦〉) | |
12 | 11 | inteqd 3641 | . . . . . . 7 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐵 = ∩ ∩ 〈𝑥, 𝑦〉) |
13 | 7, 8 | op1stb 4227 | . . . . . . 7 ⊢ ∩ ∩ 〈𝑥, 𝑦〉 = 𝑥 |
14 | 12, 13 | syl6eq 2129 | . . . . . 6 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐵 = 𝑥) |
15 | 14 | eleq1d 2147 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (∩ ∩ 𝐵 ∈ dom 𝐴 ↔ 𝑥 ∈ dom 𝐴)) |
16 | 10, 15 | sylibrd 167 | . . . 4 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴)) |
17 | 16 | exlimivv 1817 | . . 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 122 | 1 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 〈cop 3401 ∩ cint 3636 × cxp 4361 dom cdm 4363 Rel wrel 4368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-int 3637 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-dm 4373 |
This theorem is referenced by: 1stdm 5828 fundmen 6309 |
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