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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 11958 | . 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 3063 ∩ cin 3065 ⊆ wss 3066 ∅c0 3358 {csn 3522 class class class wbr 3924 × cxp 4532 dom cdm 4534 Fun wfun 5112 ‘cfv 5118 ≤ cle 7794 ℕcn 8713 ...cfz 9783 Struct cstr 11944 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-struct 11950 |
This theorem is referenced by: structcnvcnv 11964 structfung 11965 setsn0fun 11985 |
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