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Mirrors > Home > MPE Home > Th. List > mpt0 | Structured version Visualization version Unicode version |
Description: A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
mpt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 4076 | . . 3 | |
2 | eqid 2622 | . . . 4 | |
3 | 2 | fnmpt 6020 | . . 3 |
4 | 1, 3 | ax-mp 5 | . 2 |
5 | fn0 6011 | . 2 | |
6 | 4, 5 | mpbi 220 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 wral 2912 cvv 3200 c0 3915 cmpt 4729 wfn 5883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-fun 5890 df-fn 5891 |
This theorem is referenced by: oarec 7642 swrd00 13418 swrdlend 13431 repswswrd 13531 0rest 16090 grpinvfval 17460 psgnfval 17920 odfval 17952 gsumconst 18334 gsum2dlem2 18370 dprd0 18430 staffval 18847 asclfval 19334 mplcoe1 19465 mplcoe5 19468 coe1fzgsumd 19672 evl1gsumd 19721 gsumfsum 19813 pjfval 20050 mavmul0 20358 submafval 20385 mdetfval 20392 nfimdetndef 20395 mdetfval1 20396 mdet0pr 20398 madufval 20443 madugsum 20449 minmar1fval 20452 cramer0 20496 nmfval 22393 mdegfval 23822 gsumvsca1 29782 gsumvsca2 29783 esumnul 30110 esumrnmpt2 30130 sitg0 30408 mrsubfval 31405 msubfval 31421 elmsubrn 31425 mvhfval 31430 msrfval 31434 matunitlindflem1 33405 matunitlindf 33407 poimirlem28 33437 liminf0 40025 cncfiooicc 40107 itgvol0 40184 stoweidlem9 40226 sge0iunmptlemfi 40630 sge0isum 40644 lincval0 42204 |
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