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| Mirrors > Home > MPE Home > Th. List > elreldm | Structured version Visualization version GIF version | ||
| Description: The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5475). (Contributed by NM, 28-Jul-2004.) | 
| Ref | Expression | 
|---|---|
| elreldm | ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-rel 5691 | . . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
| 2 | ssel 3976 | . . . . 5 ⊢ (𝐴 ⊆ (V × V) → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) | |
| 3 | 1, 2 | sylbi 217 | . . . 4 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) | 
| 4 | elvv 5759 | . . . 4 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) | |
| 5 | 3, 4 | imbitrdi 251 | . . 3 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉)) | 
| 6 | eleq1 2828 | . . . . . 6 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
| 7 | vex 3483 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 8 | vex 3483 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 9 | 7, 8 | opeldm 5917 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) | 
| 10 | 6, 9 | biimtrdi 253 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → 𝑥 ∈ dom 𝐴)) | 
| 11 | inteq 4948 | . . . . . . . 8 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ 𝐵 = ∩ 〈𝑥, 𝑦〉) | |
| 12 | 11 | inteqd 4950 | . . . . . . 7 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐵 = ∩ ∩ 〈𝑥, 𝑦〉) | 
| 13 | 7, 8 | op1stb 5475 | . . . . . . 7 ⊢ ∩ ∩ 〈𝑥, 𝑦〉 = 𝑥 | 
| 14 | 12, 13 | eqtrdi 2792 | . . . . . 6 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐵 = 𝑥) | 
| 15 | 14 | eleq1d 2825 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (∩ ∩ 𝐵 ∈ dom 𝐴 ↔ 𝑥 ∈ dom 𝐴)) | 
| 16 | 10, 15 | sylibrd 259 | . . . 4 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴)) | 
| 17 | 16 | exlimivv 1931 | . . 3 ⊢ (∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴)) | 
| 18 | 5, 17 | syli 39 | . 2 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴)) | 
| 19 | 18 | imp 406 | 1 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 〈cop 4631 ∩ cint 4945 × cxp 5682 dom cdm 5684 Rel wrel 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-int 4946 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-dm 5694 | 
| This theorem is referenced by: 1stdm 8066 fundmen 9072 | 
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