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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 6283 | . 2 ⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V)) | |
2 | relrelss 5170 | . . . 4 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) ↔ 𝐹 ⊆ ((V × V) × V)) | |
3 | ssun1 3313 | . . . . . 6 ⊢ (V × V) ⊆ ((V × V) ∪ {∅}) | |
4 | xpss1 4751 | . . . . . 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 3178 | . . . . 5 ⊢ ((𝐹 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) | |
7 | 5, 6 | mpan2 425 | . . . 4 ⊢ (𝐹 ⊆ ((V × V) × V) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) |
8 | 2, 7 | sylbi 121 | . . 3 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) |
9 | df-ss 3157 | . . 3 ⊢ (𝐹 ⊆ (((V × V) ∪ {∅}) × V) ↔ (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹) | |
10 | 8, 9 | sylib 122 | . 2 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹) |
11 | 1, 10 | eqtrid 2234 | 1 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 Vcvv 2752 ∪ cun 3142 ∩ cin 3143 ⊆ wss 3144 ∅c0 3437 {csn 3607 × cxp 4639 dom cdm 4641 Rel wrel 4646 tpos ctpos 6263 |
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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 |
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-sbc 2978 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-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-fv 5239 df-tpos 6264 |
This theorem is referenced by: (None) |
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