Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6037 | . 2 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
2 | funss 6360 | . 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 3924 ◡ccnv 5540 Fun wfun 6335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-fun 6343 |
This theorem is referenced by: funcnvres2 6420 inpreima 6820 difpreima 6821 f1oresrab 6875 sbthlem8 8620 fin1a2lem7 9814 cnclima 21859 iscncl 21860 qtopcld 22304 qtoprest 22308 qtopcmap 22310 rnelfmlem 22543 fmfnfmlem3 22547 mbfimaicc 24215 ismbf3d 24238 i1fd 24265 gsummpt2co 30693 |
Copyright terms: Public domain | W3C validator |