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Mirrors > Home > MPE Home > Th. List > relbrtpos | Structured version Visualization version GIF version |
Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.) |
Ref | Expression |
---|---|
relbrtpos | ⊢ (Rel 𝐹 → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reltpos 7244 | . . . 4 ⊢ Rel tpos 𝐹 | |
2 | 1 | a1i 11 | . . 3 ⊢ (Rel 𝐹 → Rel tpos 𝐹) |
3 | brrelex2 5081 | . . 3 ⊢ ((Rel tpos 𝐹 ∧ 〈𝐴, 𝐵〉tpos 𝐹𝐶) → 𝐶 ∈ V) | |
4 | 2, 3 | sylan 487 | . 2 ⊢ ((Rel 𝐹 ∧ 〈𝐴, 𝐵〉tpos 𝐹𝐶) → 𝐶 ∈ V) |
5 | brrelex2 5081 | . 2 ⊢ ((Rel 𝐹 ∧ 〈𝐵, 𝐴〉𝐹𝐶) → 𝐶 ∈ V) | |
6 | brtpos 7248 | . 2 ⊢ (𝐶 ∈ V → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) | |
7 | 4, 5, 6 | pm5.21nd 939 | 1 ⊢ (Rel 𝐹 → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 Rel wrel 5043 tpos ctpos 7238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-tpos 7239 |
This theorem is referenced by: (None) |
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