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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 6606 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | funmpt2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | funeqi 6589 | . 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)) |
4 | 1, 3 | mpbir 231 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ↦ cmpt 5231 Fun wfun 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-fun 6565 |
This theorem is referenced by: pwfilem 9354 cantnfp1lem1 9716 tz9.12lem2 9826 tz9.12lem3 9827 rankf 9832 djuun 9964 cardf2 9981 fin23lem30 10380 hashf1rn 14388 oppccatf 17775 funtopon 22942 qustgpopn 24144 ustn0 24245 cphsscph 25299 ipasslem8 30866 xppreima2 32668 funcnvmpt 32684 mptiffisupp 32708 fsuppcurry1 32743 fsuppcurry2 32744 gsummpt2co 33034 zarclsint 33833 zartopn 33836 zarmxt1 33841 zarcmplem 33842 brsiga 34164 sseqval 34370 ballotlem7 34517 sinccvglem 35657 bj-evalfun 37056 bj-ccinftydisj 37196 bj-elccinfty 37197 bj-minftyccb 37208 iscard4 43523 harval3 43528 comptiunov2i 43696 icccncfext 45843 stoweidlem27 45983 stirlinglem14 46043 fourierdlem70 46132 fourierdlem71 46133 hoi2toco 46563 mptcfsupp 48222 lcoc0 48268 lincresunit2 48324 |
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