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Mirrors > Home > ILE Home > Th. List > reltpos | GIF version |
Description: The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
reltpos | ⊢ Rel tpos 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposssxp 6154 | . 2 ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | |
2 | relxp 4656 | . 2 ⊢ Rel ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | |
3 | relss 4634 | . 2 ⊢ (tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) → (Rel ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) → Rel tpos 𝐹)) | |
4 | 1, 2, 3 | mp2 16 | 1 ⊢ Rel tpos 𝐹 |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3074 ⊆ wss 3076 ∅c0 3368 {csn 3532 × cxp 4545 ◡ccnv 4546 dom cdm 4547 ran crn 4548 Rel wrel 4552 tpos ctpos 6149 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-mpt 3999 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-tpos 6150 |
This theorem is referenced by: brtpos2 6156 dftpos2 6166 dftpos3 6167 tpostpos 6169 |
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