Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4562 | . . . . . 6 ⊢ ¬ ∅ ∈ (V × V) | |
2 | cnvcnv 4986 | . . . . . . . 8 ⊢ ◡◡𝐹 = (𝐹 ∩ (V × V)) | |
3 | inss2 3292 | . . . . . . . 8 ⊢ (𝐹 ∩ (V × V)) ⊆ (V × V) | |
4 | 2, 3 | eqsstri 3124 | . . . . . . 7 ⊢ ◡◡𝐹 ⊆ (V × V) |
5 | 4 | sseli 3088 | . . . . . 6 ⊢ (∅ ∈ ◡◡𝐹 → ∅ ∈ (V × V)) |
6 | 1, 5 | mto 651 | . . . . 5 ⊢ ¬ ∅ ∈ ◡◡𝐹 |
7 | disjsn 3580 | . . . . 5 ⊢ ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡◡𝐹) | |
8 | 6, 7 | mpbir 145 | . . . 4 ⊢ (◡◡𝐹 ∩ {∅}) = ∅ |
9 | cnvcnvss 4988 | . . . . 5 ⊢ ◡◡𝐹 ⊆ 𝐹 | |
10 | reldisj 3409 | . . . . 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 11961 | . . . . 5 ⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) | |
15 | funrel 5135 | . . . . 5 ⊢ (Fun (𝐹 ∖ {∅}) → Rel (𝐹 ∖ {∅})) | |
16 | 14, 15 | syl 14 | . . . 4 ⊢ (𝐹 Struct 𝑋 → Rel (𝐹 ∖ {∅})) |
17 | dfrel2 4984 | . . . 4 ⊢ (Rel (𝐹 ∖ {∅}) ↔ ◡◡(𝐹 ∖ {∅}) = (𝐹 ∖ {∅})) | |
18 | 16, 17 | sylib 121 | . . 3 ⊢ (𝐹 Struct 𝑋 → ◡◡(𝐹 ∖ {∅}) = (𝐹 ∖ {∅})) |
19 | difss 3197 | . . . 4 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹 | |
20 | cnvss 4707 | . . . 4 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → ◡(𝐹 ∖ {∅}) ⊆ ◡𝐹) | |
21 | cnvss 4707 | . . . 4 ⊢ (◡(𝐹 ∖ {∅}) ⊆ ◡𝐹 → ◡◡(𝐹 ∖ {∅}) ⊆ ◡◡𝐹) | |
22 | 19, 20, 21 | mp2b 8 | . . 3 ⊢ ◡◡(𝐹 ∖ {∅}) ⊆ ◡◡𝐹 |
23 | 18, 22 | eqsstrrdi 3145 | . 2 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∖ {∅}) ⊆ ◡◡𝐹) |
24 | 13, 23 | eqssd 3109 | 1 ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅})) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2681 ∖ cdif 3063 ∩ cin 3065 ⊆ wss 3066 ∅c0 3358 {csn 3522 class class class wbr 3924 × cxp 4532 ◡ccnv 4533 Rel wrel 4539 Fun wfun 5112 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-in1 603 ax-in2 604 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-fal 1337 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-ne 2307 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-nul 3359 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: structfung 11965 |
Copyright terms: Public domain | W3C validator |