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Mirrors > Home > MPE Home > Th. List > mpt0 | Structured version Visualization version GIF version |
Description: A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
mpt0 | ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 4220 | . . 3 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ V | |
2 | eqid 2760 | . . . 4 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴) | |
3 | 2 | fnmpt 6181 | . . 3 ⊢ (∀𝑥 ∈ ∅ 𝐴 ∈ V → (𝑥 ∈ ∅ ↦ 𝐴) Fn ∅) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) Fn ∅ |
5 | fn0 6172 | . 2 ⊢ ((𝑥 ∈ ∅ ↦ 𝐴) Fn ∅ ↔ (𝑥 ∈ ∅ ↦ 𝐴) = ∅) | |
6 | 4, 5 | mpbi 220 | 1 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2139 ∀wral 3050 Vcvv 3340 ∅c0 4058 ↦ cmpt 4881 Fn wfn 6044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-fun 6051 df-fn 6052 |
This theorem is referenced by: oarec 7813 swrd00 13637 swrdlend 13651 repswswrd 13751 0rest 16312 grpinvfval 17681 psgnfval 18140 odfval 18172 gsumconst 18554 gsum2dlem2 18590 dprd0 18650 staffval 19069 asclfval 19556 mplcoe1 19687 mplcoe5 19690 coe1fzgsumd 19894 evl1gsumd 19943 gsumfsum 20035 pjfval 20272 mavmul0 20580 submafval 20607 mdetfval 20614 nfimdetndef 20617 mdetfval1 20618 mdet0pr 20620 madufval 20665 madugsum 20671 minmar1fval 20674 cramer0 20718 nmfval 22614 mdegfval 24041 gsumvsca1 30112 gsumvsca2 30113 esumnul 30440 esumrnmpt2 30460 sitg0 30738 mrsubfval 31733 msubfval 31749 elmsubrn 31753 mvhfval 31758 msrfval 31762 matunitlindflem1 33736 matunitlindf 33738 poimirlem28 33768 liminf0 40546 cncfiooicc 40628 itgvol0 40705 stoweidlem9 40747 sge0iunmptlemfi 41151 sge0isum 41165 lincval0 42732 |
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