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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 6604 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 2 | funmpt2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 3 | 2 | funeqi 6587 | . 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)) |
| 4 | 1, 3 | mpbir 231 | 1 ⊢ Fun 𝐹 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ↦ cmpt 5225 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 |
| This theorem is referenced by: pwfilem 9356 cantnfp1lem1 9718 tz9.12lem2 9828 tz9.12lem3 9829 rankf 9834 djuun 9966 cardf2 9983 fin23lem30 10382 hashf1rn 14391 oppccatf 17771 funtopon 22926 qustgpopn 24128 ustn0 24229 cphsscph 25285 ipasslem8 30856 xppreima2 32661 funcnvmpt 32677 mptiffisupp 32702 fsuppcurry1 32736 fsuppcurry2 32737 gsummpt2co 33051 zarclsint 33871 zartopn 33874 zarmxt1 33879 zarcmplem 33880 brsiga 34184 sseqval 34390 ballotlem7 34538 sinccvglem 35677 bj-evalfun 37074 bj-ccinftydisj 37214 bj-elccinfty 37215 bj-minftyccb 37226 iscard4 43546 harval3 43551 comptiunov2i 43719 icccncfext 45902 stoweidlem27 46042 stirlinglem14 46102 fourierdlem70 46191 fourierdlem71 46192 hoi2toco 46622 mptcfsupp 48293 lcoc0 48339 lincresunit2 48395 |
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