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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 8237 | . 2 ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | |
2 | relxp 5692 | . 2 ⊢ Rel ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | |
3 | relss 5779 | . 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 3944 ⊆ wss 3946 ∅c0 4322 {csn 4623 × cxp 5672 ◡ccnv 5673 dom cdm 5674 ran crn 5675 Rel wrel 5679 tpos ctpos 8232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5146 df-opab 5208 df-mpt 5229 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-tpos 8233 |
This theorem is referenced by: brtpos2 8239 relbrtpos 8244 dftpos2 8250 dftpos3 8251 tpostpos 8253 |
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