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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 6097 | . 2 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
2 | funss 6453 | . 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 3887 ◡ccnv 5588 Fun wfun 6427 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-fun 6435 |
This theorem is referenced by: funcnvres2 6514 inpreima 6941 difpreima 6942 f1oresrab 6999 sbthlem8 8877 fin1a2lem7 10162 cnclima 22419 iscncl 22420 qtopcld 22864 qtoprest 22868 qtopcmap 22870 rnelfmlem 23103 fmfnfmlem3 23107 mbfimaicc 24795 ismbf3d 24818 i1fd 24845 gsummpt2co 31308 |
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