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Mirrors > Home > ILE Home > Th. List > dmsnop | GIF version |
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
dmsnop.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
dmsnop | ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmsnop.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | dmsnopg 5018 | . 2 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 Vcvv 2689 {csn 3532 〈cop 3535 dom cdm 4547 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-dm 4557 |
This theorem is referenced by: dmtpop 5022 dmsnsnsng 5024 op1sta 5028 funtp 5184 ac6sfi 6800 |
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