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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 5803 | . 2 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
2 | funss 6118 | . 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 3767 ◡ccnv 5309 Fun wfun 6093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-br 4842 df-opab 4904 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-fun 6101 |
This theorem is referenced by: funcnvres2 6178 inpreima 6566 difpreima 6567 f1oresrab 6619 sbthlem8 8317 fin1a2lem7 9514 cnclima 21398 iscncl 21399 qtopcld 21842 qtoprest 21846 qtopcmap 21848 rnelfmlem 22081 fmfnfmlem3 22085 mbfimaicc 23736 ismbf3d 23759 i1fd 23786 gsummpt2co 30288 |
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