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Mirrors > Home > MPE Home > Th. List > funcnvcnv | Structured version Visualization version GIF version |
Description: The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.) |
Ref | Expression |
---|---|
funcnvcnv | ⊢ (Fun 𝐴 → Fun ◡◡𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnvss 5507 | . 2 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
2 | funss 5822 | . 2 ⊢ (◡◡𝐴 ⊆ 𝐴 → (Fun 𝐴 → Fun ◡◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Fun 𝐴 → Fun ◡◡𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3540 ◡ccnv 5037 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-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-fun 5806 |
This theorem is referenced by: funcnvres2 5883 inpreima 6250 difpreima 6251 f1oresrab 6302 sbthlem8 7962 fin1a2lem7 9111 strlemor0 15795 cnclima 20882 iscncl 20883 qtopcld 21326 qtoprest 21330 qtopcmap 21332 rnelfmlem 21566 fmfnfmlem3 21570 mbfimaicc 23206 ismbf3d 23227 i1fd 23254 gsummpt2co 29111 |
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