![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5429). (Contributed by NM, 28-Jul-2004.) |
Ref | Expression |
---|---|
elreldm | ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5641 | . . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
2 | ssel 3938 | . . . . 5 ⊢ (𝐴 ⊆ (V × V) → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) | |
3 | 1, 2 | sylbi 216 | . . . 4 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) |
4 | elvv 5707 | . . . 4 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩) | |
5 | 3, 4 | syl6ib 251 | . . 3 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)) |
6 | eleq1 2822 | . . . . . 6 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)) | |
7 | vex 3448 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | vex 3448 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opeldm 5864 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
10 | 6, 9 | syl6bi 253 | . . . . 5 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 → 𝑥 ∈ dom 𝐴)) |
11 | inteq 4911 | . . . . . . . 8 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ 𝐵 = ∩ ⟨𝑥, 𝑦⟩) | |
12 | 11 | inteqd 4913 | . . . . . . 7 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐵 = ∩ ∩ ⟨𝑥, 𝑦⟩) |
13 | 7, 8 | op1stb 5429 | . . . . . . 7 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥 |
14 | 12, 13 | eqtrdi 2789 | . . . . . 6 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐵 = 𝑥) |
15 | 14 | eleq1d 2819 | . . . . 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 1936 | . . 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 408 | 1 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 ⟨cop 4593 ∩ cint 4908 × cxp 5632 dom cdm 5634 Rel wrel 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-int 4909 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-dm 5644 |
This theorem is referenced by: 1stdm 7973 fundmen 8978 |
Copyright terms: Public domain | W3C validator |