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Mirrors > Home > ILE Home > Th. List > tpostpos2 | GIF version |
Description: Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
tpostpos2 | ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpostpos 6154 | . 2 ⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V)) | |
2 | relrelss 5060 | . . . 4 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) ↔ 𝐹 ⊆ ((V × V) × V)) | |
3 | ssun1 3234 | . . . . . 6 ⊢ (V × V) ⊆ ((V × V) ∪ {∅}) | |
4 | xpss1 4644 | . . . . . 6 ⊢ ((V × V) ⊆ ((V × V) ∪ {∅}) → ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V) |
6 | sstr 3100 | . . . . 5 ⊢ ((𝐹 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) | |
7 | 5, 6 | mpan2 421 | . . . 4 ⊢ (𝐹 ⊆ ((V × V) × V) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) |
8 | 2, 7 | sylbi 120 | . . 3 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) |
9 | df-ss 3079 | . . 3 ⊢ (𝐹 ⊆ (((V × V) ∪ {∅}) × V) ↔ (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹) | |
10 | 8, 9 | sylib 121 | . 2 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹) |
11 | 1, 10 | syl5eq 2182 | 1 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 Vcvv 2681 ∪ cun 3064 ∩ cin 3065 ⊆ wss 3066 ∅c0 3358 {csn 3522 × cxp 4532 dom cdm 4534 Rel wrel 4539 tpos ctpos 6134 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
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-sbc 2905 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-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 df-tpos 6135 |
This theorem is referenced by: (None) |
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