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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 4626 | . . . . . 6 ⊢ ¬ ∅ ∈ (V × V) | |
2 | cnvcnv 5050 | . . . . . . . 8 ⊢ ◡◡𝐹 = (𝐹 ∩ (V × V)) | |
3 | inss2 3338 | . . . . . . . 8 ⊢ (𝐹 ∩ (V × V)) ⊆ (V × V) | |
4 | 2, 3 | eqsstri 3169 | . . . . . . 7 ⊢ ◡◡𝐹 ⊆ (V × V) |
5 | 4 | sseli 3133 | . . . . . 6 ⊢ (∅ ∈ ◡◡𝐹 → ∅ ∈ (V × V)) |
6 | 1, 5 | mto 652 | . . . . 5 ⊢ ¬ ∅ ∈ ◡◡𝐹 |
7 | disjsn 3632 | . . . . 5 ⊢ ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡◡𝐹) | |
8 | 6, 7 | mpbir 145 | . . . 4 ⊢ (◡◡𝐹 ∩ {∅}) = ∅ |
9 | cnvcnvss 5052 | . . . . 5 ⊢ ◡◡𝐹 ⊆ 𝐹 | |
10 | reldisj 3455 | . . . . 5 ⊢ (◡◡𝐹 ⊆ 𝐹 → ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ◡◡𝐹 ⊆ (𝐹 ∖ {∅}))) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ◡◡𝐹 ⊆ (𝐹 ∖ {∅})) |
12 | 8, 11 | mpbi 144 | . . 3 ⊢ ◡◡𝐹 ⊆ (𝐹 ∖ {∅}) |
13 | 12 | a1i 9 | . 2 ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 ⊆ (𝐹 ∖ {∅})) |
14 | structn0fun 12350 | . . . . 5 ⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) | |
15 | funrel 5199 | . . . . 5 ⊢ (Fun (𝐹 ∖ {∅}) → Rel (𝐹 ∖ {∅})) | |
16 | 14, 15 | syl 14 | . . . 4 ⊢ (𝐹 Struct 𝑋 → Rel (𝐹 ∖ {∅})) |
17 | dfrel2 5048 | . . . 4 ⊢ (Rel (𝐹 ∖ {∅}) ↔ ◡◡(𝐹 ∖ {∅}) = (𝐹 ∖ {∅})) | |
18 | 16, 17 | sylib 121 | . . 3 ⊢ (𝐹 Struct 𝑋 → ◡◡(𝐹 ∖ {∅}) = (𝐹 ∖ {∅})) |
19 | difss 3243 | . . . 4 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹 | |
20 | cnvss 4771 | . . . 4 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → ◡(𝐹 ∖ {∅}) ⊆ ◡𝐹) | |
21 | cnvss 4771 | . . . 4 ⊢ (◡(𝐹 ∖ {∅}) ⊆ ◡𝐹 → ◡◡(𝐹 ∖ {∅}) ⊆ ◡◡𝐹) | |
22 | 19, 20, 21 | mp2b 8 | . . 3 ⊢ ◡◡(𝐹 ∖ {∅}) ⊆ ◡◡𝐹 |
23 | 18, 22 | eqsstrrdi 3190 | . 2 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∖ {∅}) ⊆ ◡◡𝐹) |
24 | 13, 23 | eqssd 3154 | 1 ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅})) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∖ cdif 3108 ∩ cin 3110 ⊆ wss 3111 ∅c0 3404 {csn 3570 class class class wbr 3976 × cxp 4596 ◡ccnv 4597 Rel wrel 4603 Fun wfun 5176 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-in1 604 ax-in2 605 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-fal 1348 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-ne 2335 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-nul 3405 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: structfung 12354 |
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