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Mirrors > Home > MPE Home > Th. List > funcnvcnv | Structured version Visualization version Unicode version |
Description: The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.) |
Ref | Expression |
---|---|
funcnvcnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnvss 5589 | . 2 | |
2 | funss 5907 | . 2 | |
3 | 1, 2 | ax-mp 5 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wss 3574 ccnv 5113 wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-fun 5890 |
This theorem is referenced by: funcnvres2 5969 inpreima 6342 difpreima 6343 f1oresrab 6395 sbthlem8 8077 fin1a2lem7 9228 strlemor0OLD 15968 cnclima 21072 iscncl 21073 qtopcld 21516 qtoprest 21520 qtopcmap 21522 rnelfmlem 21756 fmfnfmlem3 21760 mbfimaicc 23400 ismbf3d 23421 i1fd 23448 gsummpt2co 29780 |
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