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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 6235 | . 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 1540 ∈ wcel 2108 Vcvv 3480 {cpr 4628 〈cop 4632 dom cdm 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 | 
| This theorem is referenced by: dmtpop 6238 funtp 6623 fpr 7174 fnprb 7228 hashfun 14476 umgr2v2evd2 29545 ex-dm 30458 | 
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