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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 5469). (Contributed by NM, 28-Jul-2004.) |
Ref | Expression |
---|---|
elreldm | ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5681 | . . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
2 | ssel 3972 | . . . . 5 ⊢ (𝐴 ⊆ (V × V) → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) | |
3 | 1, 2 | sylbi 216 | . . . 4 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) |
4 | elvv 5748 | . . . 4 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) | |
5 | 3, 4 | imbitrdi 250 | . . 3 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉)) |
6 | eleq1 2814 | . . . . . 6 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
7 | vex 3466 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | vex 3466 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opeldm 5906 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
10 | 6, 9 | biimtrdi 252 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → 𝑥 ∈ dom 𝐴)) |
11 | inteq 4949 | . . . . . . . 8 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ 𝐵 = ∩ 〈𝑥, 𝑦〉) | |
12 | 11 | inteqd 4951 | . . . . . . 7 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐵 = ∩ ∩ 〈𝑥, 𝑦〉) |
13 | 7, 8 | op1stb 5469 | . . . . . . 7 ⊢ ∩ ∩ 〈𝑥, 𝑦〉 = 𝑥 |
14 | 12, 13 | eqtrdi 2782 | . . . . . 6 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐵 = 𝑥) |
15 | 14 | eleq1d 2811 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (∩ ∩ 𝐵 ∈ dom 𝐴 ↔ 𝑥 ∈ dom 𝐴)) |
16 | 10, 15 | sylibrd 258 | . . . 4 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴)) |
17 | 16 | exlimivv 1928 | . . 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 405 | 1 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3946 〈cop 4629 ∩ cint 4946 × cxp 5672 dom cdm 5674 Rel wrel 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-int 4947 df-br 5146 df-opab 5208 df-xp 5680 df-rel 5681 df-dm 5684 |
This theorem is referenced by: 1stdm 8046 fundmen 9059 |
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