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Mirrors > Home > MPE Home > Th. List > funmpt2 | Structured version Visualization version GIF version |
Description: Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.) |
Ref | Expression |
---|---|
funmpt2.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
funmpt2 | ⊢ Fun 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 6087 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | funmpt2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | funeqi 6070 | . 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)) |
4 | 1, 3 | mpbir 221 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ↦ cmpt 4881 Fun wfun 6043 |
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-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-fun 6051 |
This theorem is referenced by: cantnfp1lem1 8750 tz9.12lem2 8826 tz9.12lem3 8827 rankf 8832 djuun 8962 cardf2 8979 fin23lem30 9376 hashf1rn 13355 funtopon 20947 qustgpopn 22144 ustn0 22245 metuval 22575 ipasslem8 28022 xppreima2 29780 funcnvmpt 29798 gsummpt2co 30110 metidval 30263 pstmval 30268 brsiga 30576 measbasedom 30595 sseqval 30780 ballotlem7 30927 sinccvglem 31894 bj-evalfun 33349 bj-ccinftydisj 33429 bj-elccinfty 33430 bj-minftyccb 33441 comptiunov2i 38518 icccncfext 40621 stoweidlem27 40765 stirlinglem14 40825 fourierdlem70 40914 fourierdlem71 40915 hoi2toco 41345 mptcfsupp 42689 lcoc0 42739 lincresunit2 42795 |
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