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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 6237 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶}) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 {cpr 4633 〈cop 4637 dom cdm 5689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-dm 5699 |
This theorem is referenced by: dmtpop 6240 funtp 6625 fpr 7174 fnprb 7228 hashfun 14473 umgr2v2evd2 29560 ex-dm 30468 |
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