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Mirrors > Home > MPE Home > Th. List > funmpt | Structured version Visualization version GIF version |
Description: A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
Ref | Expression |
---|---|
funmpt | ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funopab4 5839 | . 2 ⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | df-mpt 4645 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
3 | 2 | funeqi 5824 | . 2 ⊢ (Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}) |
4 | 1, 3 | mpbir 220 | 1 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 {copab 4642 ↦ cmpt 4643 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-fun 5806 |
This theorem is referenced by: funmpt2 5841 resfunexg 6384 mptexg 6389 brtpos2 7245 tposfun 7255 mptfi 8148 sniffsupp 8198 cantnfrescl 8456 cantnflem1 8469 r0weon 8718 axcc2lem 9141 negfi 10850 mptnn0fsupp 12659 ccatalpha 13228 mreacs 16142 acsfn 16143 isofval 16240 lubfun 16803 glbfun 16816 acsficl2d 16999 pmtrsn 17762 gsum2dlem2 18193 gsum2d 18194 dprdfinv 18241 dprdfadd 18242 dmdprdsplitlem 18259 dpjidcl 18280 mptscmfsupp0 18751 00lsp 18802 psrass1lem 19198 psrlidm 19224 psrridm 19225 psrass1 19226 psrass23l 19229 psrcom 19230 psrass23 19231 mplsubrg 19261 mplmon 19284 mplmonmul 19285 mplcoe1 19286 mplcoe5 19289 mplbas2 19291 evlslem2 19333 evlslem6 19334 psropprmul 19429 coe1mul2 19460 pjpm 19871 frlmphllem 19938 frlmphl 19939 uvcff 19949 uvcresum 19951 oftpos 20077 pmatcollpw2lem 20401 tgrest 20773 cmpfi 21021 1stcrestlem 21065 ptcnplem 21234 xkoinjcn 21300 symgtgp 21715 eltsms 21746 rrxmval 22996 tdeglem4 23624 plypf1 23772 tayl0 23920 taylthlem1 23931 xrlimcnp 24495 abrexexd 28731 mptexgf 28809 fmptcof2 28839 ofpreima 28848 funcnvmptOLD 28850 mptct 28880 mptctf 28883 psgnfzto1stlem 29181 locfinreflem 29235 measdivcstOLD 29614 sxbrsigalem0 29660 sitgf 29736 imageval 31207 poimirlem30 32609 poimir 32612 choicefi 38387 fourierdlem80 39079 sge0tsms 39273 scmsuppss 41947 rmfsupp 41949 scmfsupp 41953 fdivval 42131 |
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