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Mirrors > Home > ILE Home > Th. List > structcnvcnv | GIF version |
Description: Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
structcnvcnv | ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 4505 | . . . . . 6 ⊢ ¬ ∅ ∈ (V × V) | |
2 | cnvcnv 4927 | . . . . . . . 8 ⊢ ◡◡𝐹 = (𝐹 ∩ (V × V)) | |
3 | inss2 3244 | . . . . . . . 8 ⊢ (𝐹 ∩ (V × V)) ⊆ (V × V) | |
4 | 2, 3 | eqsstri 3079 | . . . . . . 7 ⊢ ◡◡𝐹 ⊆ (V × V) |
5 | 4 | sseli 3043 | . . . . . 6 ⊢ (∅ ∈ ◡◡𝐹 → ∅ ∈ (V × V)) |
6 | 1, 5 | mto 629 | . . . . 5 ⊢ ¬ ∅ ∈ ◡◡𝐹 |
7 | disjsn 3532 | . . . . 5 ⊢ ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡◡𝐹) | |
8 | 6, 7 | mpbir 145 | . . . 4 ⊢ (◡◡𝐹 ∩ {∅}) = ∅ |
9 | cnvcnvss 4929 | . . . . 5 ⊢ ◡◡𝐹 ⊆ 𝐹 | |
10 | reldisj 3361 | . . . . 5 ⊢ (◡◡𝐹 ⊆ 𝐹 → ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ◡◡𝐹 ⊆ (𝐹 ∖ {∅}))) | |
11 | 9, 10 | ax-mp 7 | . . . 4 ⊢ ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ◡◡𝐹 ⊆ (𝐹 ∖ {∅})) |
12 | 8, 11 | mpbi 144 | . . 3 ⊢ ◡◡𝐹 ⊆ (𝐹 ∖ {∅}) |
13 | 12 | a1i 9 | . 2 ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 ⊆ (𝐹 ∖ {∅})) |
14 | structn0fun 11754 | . . . . 5 ⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) | |
15 | funrel 5076 | . . . . 5 ⊢ (Fun (𝐹 ∖ {∅}) → Rel (𝐹 ∖ {∅})) | |
16 | 14, 15 | syl 14 | . . . 4 ⊢ (𝐹 Struct 𝑋 → Rel (𝐹 ∖ {∅})) |
17 | dfrel2 4925 | . . . 4 ⊢ (Rel (𝐹 ∖ {∅}) ↔ ◡◡(𝐹 ∖ {∅}) = (𝐹 ∖ {∅})) | |
18 | 16, 17 | sylib 121 | . . 3 ⊢ (𝐹 Struct 𝑋 → ◡◡(𝐹 ∖ {∅}) = (𝐹 ∖ {∅})) |
19 | difss 3149 | . . . 4 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹 | |
20 | cnvss 4650 | . . . 4 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → ◡(𝐹 ∖ {∅}) ⊆ ◡𝐹) | |
21 | cnvss 4650 | . . . 4 ⊢ (◡(𝐹 ∖ {∅}) ⊆ ◡𝐹 → ◡◡(𝐹 ∖ {∅}) ⊆ ◡◡𝐹) | |
22 | 19, 20, 21 | mp2b 8 | . . 3 ⊢ ◡◡(𝐹 ∖ {∅}) ⊆ ◡◡𝐹 |
23 | 18, 22 | syl6eqssr 3100 | . 2 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∖ {∅}) ⊆ ◡◡𝐹) |
24 | 13, 23 | eqssd 3064 | 1 ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅})) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1299 ∈ wcel 1448 Vcvv 2641 ∖ cdif 3018 ∩ cin 3020 ⊆ wss 3021 ∅c0 3310 {csn 3474 class class class wbr 3875 × cxp 4475 ◡ccnv 4476 Rel wrel 4482 Fun wfun 5053 Struct cstr 11737 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-struct 11743 |
This theorem is referenced by: structfung 11758 |
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