Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > breldm | GIF version |
Description: Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.) |
Ref | Expression |
---|---|
opeldm.1 | ⊢ 𝐴 ∈ V |
opeldm.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
breldm | ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3900 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | opeldm.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | opeldm.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | opeldm 4712 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 → 𝐴 ∈ dom 𝑅) |
5 | 1, 4 | sylbi 120 | 1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 Vcvv 2660 〈cop 3500 class class class wbr 3899 dom cdm 4509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-dm 4519 |
This theorem is referenced by: exse2 4883 funcnv3 5155 dff13 5637 reldmtpos 6118 rntpos 6122 dftpos4 6128 tpostpos 6129 iserd 6423 |
Copyright terms: Public domain | W3C validator |