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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 6046 | . 2 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
2 | funss 6369 | . 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 3936 ◡ccnv 5549 Fun wfun 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-fun 6352 |
This theorem is referenced by: funcnvres2 6429 inpreima 6829 difpreima 6830 f1oresrab 6884 sbthlem8 8628 fin1a2lem7 9822 cnclima 21870 iscncl 21871 qtopcld 22315 qtoprest 22319 qtopcmap 22321 rnelfmlem 22554 fmfnfmlem3 22558 mbfimaicc 24226 ismbf3d 24249 i1fd 24276 gsummpt2co 30681 |
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