Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > structn0fun | GIF version |
Description: A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.) |
Ref | Expression |
---|---|
structn0fun | ⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isstruct2im 11979 | . 2 ⊢ (𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))) | |
2 | 1 | simp2d 994 | 1 ⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∖ cdif 3068 ∩ cin 3070 ⊆ wss 3071 ∅c0 3363 {csn 3527 class class class wbr 3929 × cxp 4537 dom cdm 4539 Fun wfun 5117 ‘cfv 5123 ≤ cle 7808 ℕcn 8727 ...cfz 9797 Struct cstr 11965 |
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-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-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 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-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 11971 |
This theorem is referenced by: structcnvcnv 11985 structfung 11986 setsn0fun 12006 |
Copyright terms: Public domain | W3C validator |