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Mirrors > Home > MPE Home > Th. List > dmprop | Structured version Visualization version 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 5526 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶}) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 {cpr 4127 〈cop 4131 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-dm 5048 |
This theorem is referenced by: dmtpop 5529 funtp 5859 fpr 6326 fnprb 6377 hashfun 13084 ex-dm 26688 umgr2v2evd2 40743 |
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