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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 5840 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | funmpt2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | funeqi 5824 | . 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)) |
4 | 1, 3 | mpbir 220 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ↦ 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: cantnfp1lem1 8458 tz9.12lem2 8534 tz9.12lem3 8535 rankf 8540 cardf2 8652 fin23lem30 9047 hashf1rn 13004 hashf1rnOLD 13005 qustgpopn 21733 ustn0 21834 metuval 22164 ipasslem8 27076 xppreima2 28830 funcnvmpt 28851 gsummpt2co 29111 metidval 29261 pstmval 29266 brsiga 29573 measbasedom 29592 sseqval 29777 ballotlem7 29924 sinccvglem 30820 bj-funtopon 32236 bj-ccinftydisj 32277 bj-elccinfty 32278 bj-minftyccb 32289 comptiunov2i 37017 icccncfext 38773 stoweidlem27 38920 stirlinglem14 38980 fourierdlem70 39069 fourierdlem71 39070 hoi2toco 39497 mptcfsupp 41955 lcoc0 42005 lincresunit2 42061 |
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