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Mirrors > Home > ILE Home > Th. List > structfung | GIF version |
Description: The converse of the converse of a structure is a function. Closed form of structfun 11980. (Contributed by AV, 12-Nov-2021.) |
Ref | Expression |
---|---|
structfung | ⊢ (𝐹 Struct 𝑋 → Fun ◡◡𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | structn0fun 11975 | . 2 ⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) | |
2 | structcnvcnv 11978 | . . 3 ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅})) | |
3 | 2 | funeqd 5145 | . 2 ⊢ (𝐹 Struct 𝑋 → (Fun ◡◡𝐹 ↔ Fun (𝐹 ∖ {∅}))) |
4 | 1, 3 | mpbird 166 | 1 ⊢ (𝐹 Struct 𝑋 → Fun ◡◡𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∖ cdif 3068 ∅c0 3363 {csn 3527 class class class wbr 3929 ◡ccnv 4538 Fun wfun 5117 Struct cstr 11958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-struct 11964 |
This theorem is referenced by: structfun 11980 opelstrsl 12058 |
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