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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 6073 | . 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 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 〈cop 4576 dom cdm 5558 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-dm 5568 |
This theorem is referenced by: dmtpop 6078 dmsnsnsn 6080 op1sta 6085 funtp 6414 funopdmsn 6915 wfrlem13 7970 wfrlem16 7973 tfrlem10 8026 ac6sfi 8765 dcomex 9872 axdc3lem4 9878 cnfldfun 20560 bnj1416 32315 bnj1421 32318 subfacp1lem2a 32431 subfacp1lem5 32435 frrlem14 33140 noextend 33177 nosupbday 33209 nosupbnd1 33218 nosupbnd2 33220 |
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