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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 6213 | . 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 1542 ∈ wcel 2107 Vcvv 3475 {csn 4629 ⟨cop 4635 dom cdm 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-dm 5687 |
This theorem is referenced by: dmtpop 6218 dmsnsnsn 6220 op1sta 6225 snres0 6298 funtp 6606 funopdmsn 7148 frrlem14 8284 wfrlem13OLD 8321 wfrlem16OLD 8324 tfrlem10 8387 ac6sfi 9287 dcomex 10442 axdc3lem4 10448 cnfldfunALT 20957 cnfldfunALTOLD 20958 noextend 27169 nosupbday 27208 nosupbnd1 27217 nosupbnd2 27219 noinfbday 27223 noinfbnd1 27232 noinfbnd2 27234 bnj1416 34050 bnj1421 34053 fineqvac 34097 subfacp1lem2a 34171 subfacp1lem5 34175 |
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