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Mirrors > Home > MPE Home > Th. List > reltpos | Structured version Visualization version 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 7890 | . 2 ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | |
2 | relxp 5567 | . 2 ⊢ Rel ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | |
3 | relss 5650 | . 2 ⊢ (tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) → (Rel ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) → Rel tpos 𝐹)) | |
4 | 1, 2, 3 | mp2 9 | 1 ⊢ Rel tpos 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 {csn 4560 × cxp 5547 ◡ccnv 5548 dom cdm 5549 ran crn 5550 Rel wrel 5554 tpos ctpos 7885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-mpt 5139 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-tpos 7886 |
This theorem is referenced by: brtpos2 7892 relbrtpos 7897 dftpos2 7903 dftpos3 7904 tpostpos 7906 |
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