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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 12347 | . 2 ⊢ (𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))) | |
2 | 1 | simp2d 999 | 1 ⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 ∖ cdif 3108 ∩ cin 3110 ⊆ wss 3111 ∅c0 3404 {csn 3570 class class class wbr 3976 × cxp 4596 dom cdm 4598 Fun wfun 5176 ‘cfv 5182 ≤ cle 7925 ℕcn 8848 ...cfz 9935 Struct cstr 12333 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-struct 12339 |
This theorem is referenced by: structcnvcnv 12353 structfung 12354 setsn0fun 12374 |
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