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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. (Contributed by NM, 28-Jul-2004.) |
Ref | Expression |
---|---|
elreldm | ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5450 | . . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
2 | ssel 3883 | . . . . 5 ⊢ (𝐴 ⊆ (V × V) → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) | |
3 | 1, 2 | sylbi 218 | . . . 4 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V))) |
4 | elvv 5512 | . . . 4 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉) | |
5 | 3, 4 | syl6ib 252 | . . 3 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∃𝑥∃𝑦 𝐵 = 〈𝑥, 𝑦〉)) |
6 | eleq1 2870 | . . . . . 6 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
7 | vex 3440 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | vex 3440 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opeldm 5662 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
10 | 6, 9 | syl6bi 254 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → 𝑥 ∈ dom 𝐴)) |
11 | inteq 4785 | . . . . . . . 8 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ 𝐵 = ∩ 〈𝑥, 𝑦〉) | |
12 | 11 | inteqd 4787 | . . . . . . 7 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐵 = ∩ ∩ 〈𝑥, 𝑦〉) |
13 | 7, 8 | op1stb 5255 | . . . . . . 7 ⊢ ∩ ∩ 〈𝑥, 𝑦〉 = 𝑥 |
14 | 12, 13 | syl6eq 2847 | . . . . . 6 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐵 = 𝑥) |
15 | 14 | eleq1d 2867 | . . . . 5 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (∩ ∩ 𝐵 ∈ dom 𝐴 ↔ 𝑥 ∈ dom 𝐴)) |
16 | 10, 15 | sylibrd 260 | . . . 4 ⊢ (𝐵 = 〈𝑥, 𝑦〉 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴)) |
17 | 16 | exlimivv 1910 | . . 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 407 | 1 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∃wex 1761 ∈ wcel 2081 Vcvv 3437 ⊆ wss 3859 〈cop 4478 ∩ cint 4782 × cxp 5441 dom cdm 5443 Rel wrel 5448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-int 4783 df-br 4963 df-opab 5025 df-xp 5449 df-rel 5450 df-dm 5453 |
This theorem is referenced by: 1stdm 7595 fundmen 8431 |
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