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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 6214 | . 2 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
| 2 | funss 6585 | . 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 3951 ◡ccnv 5684 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-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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 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-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 |
| This theorem is referenced by: funcnvres2 6646 inpreima 7084 difpreima 7085 f1oresrab 7147 sbthlem8 9130 fin1a2lem7 10446 cnclima 23276 iscncl 23277 qtopcld 23721 qtoprest 23725 qtopcmap 23727 rnelfmlem 23960 fmfnfmlem3 23964 mbfimaicc 25666 ismbf3d 25689 i1fd 25716 gsummpt2co 33051 |
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