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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 4669 | . . . . . 6 ⊢ ¬ ∅ ∈ (V × V) | |
2 | cnvcnv 5096 | . . . . . . . 8 ⊢ ◡◡𝐹 = (𝐹 ∩ (V × V)) | |
3 | inss2 3371 | . . . . . . . 8 ⊢ (𝐹 ∩ (V × V)) ⊆ (V × V) | |
4 | 2, 3 | eqsstri 3202 | . . . . . . 7 ⊢ ◡◡𝐹 ⊆ (V × V) |
5 | 4 | sseli 3166 | . . . . . 6 ⊢ (∅ ∈ ◡◡𝐹 → ∅ ∈ (V × V)) |
6 | 1, 5 | mto 663 | . . . . 5 ⊢ ¬ ∅ ∈ ◡◡𝐹 |
7 | disjsn 3669 | . . . . 5 ⊢ ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡◡𝐹) | |
8 | 6, 7 | mpbir 146 | . . . 4 ⊢ (◡◡𝐹 ∩ {∅}) = ∅ |
9 | cnvcnvss 5098 | . . . . 5 ⊢ ◡◡𝐹 ⊆ 𝐹 | |
10 | reldisj 3489 | . . . . 5 ⊢ (◡◡𝐹 ⊆ 𝐹 → ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ◡◡𝐹 ⊆ (𝐹 ∖ {∅}))) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ◡◡𝐹 ⊆ (𝐹 ∖ {∅})) |
12 | 8, 11 | mpbi 145 | . . 3 ⊢ ◡◡𝐹 ⊆ (𝐹 ∖ {∅}) |
13 | 12 | a1i 9 | . 2 ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 ⊆ (𝐹 ∖ {∅})) |
14 | structn0fun 12500 | . . . . 5 ⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) | |
15 | funrel 5249 | . . . . 5 ⊢ (Fun (𝐹 ∖ {∅}) → Rel (𝐹 ∖ {∅})) | |
16 | 14, 15 | syl 14 | . . . 4 ⊢ (𝐹 Struct 𝑋 → Rel (𝐹 ∖ {∅})) |
17 | dfrel2 5094 | . . . 4 ⊢ (Rel (𝐹 ∖ {∅}) ↔ ◡◡(𝐹 ∖ {∅}) = (𝐹 ∖ {∅})) | |
18 | 16, 17 | sylib 122 | . . 3 ⊢ (𝐹 Struct 𝑋 → ◡◡(𝐹 ∖ {∅}) = (𝐹 ∖ {∅})) |
19 | difss 3276 | . . . 4 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹 | |
20 | cnvss 4815 | . . . 4 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → ◡(𝐹 ∖ {∅}) ⊆ ◡𝐹) | |
21 | cnvss 4815 | . . . 4 ⊢ (◡(𝐹 ∖ {∅}) ⊆ ◡𝐹 → ◡◡(𝐹 ∖ {∅}) ⊆ ◡◡𝐹) | |
22 | 19, 20, 21 | mp2b 8 | . . 3 ⊢ ◡◡(𝐹 ∖ {∅}) ⊆ ◡◡𝐹 |
23 | 18, 22 | eqsstrrdi 3223 | . 2 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∖ {∅}) ⊆ ◡◡𝐹) |
24 | 13, 23 | eqssd 3187 | 1 ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅})) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ∖ cdif 3141 ∩ cin 3143 ⊆ wss 3144 ∅c0 3437 {csn 3607 class class class wbr 4018 × cxp 4639 ◡ccnv 4640 Rel wrel 4646 Fun wfun 5226 Struct cstr 12483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5234 df-fv 5240 df-struct 12489 |
This theorem is referenced by: structfung 12504 |
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