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Mirrors > Home > ILE Home > Th. List > dmprop | GIF version |
Description: The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
dmsnop.1 | ⊢ 𝐵 ∈ V |
dmprop.1 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
dmprop | ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmsnop.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | dmprop.1 | . 2 ⊢ 𝐷 ∈ V | |
3 | dmpropg 5101 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶}) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶} |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 Vcvv 2737 {cpr 3593 〈cop 3595 dom cdm 4626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-dm 4636 |
This theorem is referenced by: dmtpop 5104 funtp 5269 fpr 5698 |
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