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| Mirrors > Home > MPE Home > Th. List > dmsnop | Structured version Visualization version 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 6233 | . 2 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈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-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-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 dmsnsnsn 6240 op1sta 6245 snres0 6318 funtp 6623 funopdmsn 7170 frrlem14 8324 wfrlem13OLD 8361 wfrlem16OLD 8364 tfrlem10 8427 ac6sfi 9320 dcomex 10487 axdc3lem4 10493 cnfldfunALT 21379 cnfldfunALTOLD 21392 cnfldfunALTOLDOLD 21393 noextend 27711 nosupbday 27750 nosupbnd1 27759 nosupbnd2 27761 noinfbday 27765 noinfbnd1 27774 noinfbnd2 27776 bnj1416 35053 bnj1421 35056 fineqvac 35111 subfacp1lem2a 35185 subfacp1lem5 35189 |
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