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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 6540 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | funmpt2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | funeqi 6523 | . 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)) |
4 | 1, 3 | mpbir 230 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ↦ cmpt 5189 Fun wfun 6491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-fun 6499 |
This theorem is referenced by: pwfilem 9124 cantnfp1lem1 9619 tz9.12lem2 9729 tz9.12lem3 9730 rankf 9735 djuun 9867 cardf2 9884 fin23lem30 10283 hashf1rn 14258 oppccatf 17615 funtopon 22285 qustgpopn 23487 ustn0 23588 cphsscph 24631 ipasslem8 29821 xppreima2 31613 funcnvmpt 31629 mptiffisupp 31654 fsuppcurry1 31689 fsuppcurry2 31690 gsummpt2co 31939 zarclsint 32510 zartopn 32513 zarmxt1 32518 zarcmplem 32519 brsiga 32839 sseqval 33045 ballotlem7 33192 sinccvglem 34317 bj-evalfun 35590 bj-ccinftydisj 35730 bj-elccinfty 35731 bj-minftyccb 35742 iscard4 41893 harval3 41898 comptiunov2i 42066 icccncfext 44214 stoweidlem27 44354 stirlinglem14 44414 fourierdlem70 44503 fourierdlem71 44504 hoi2toco 44934 mptcfsupp 46542 lcoc0 46589 lincresunit2 46645 |
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