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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 7897 | . . . 4 ⊢ Rel tpos 𝐹 | |
2 | 1 | a1i 11 | . . 3 ⊢ (Rel 𝐹 → Rel tpos 𝐹) |
3 | brrelex2 5606 | . . 3 ⊢ ((Rel tpos 𝐹 ∧ 〈𝐴, 𝐵〉tpos 𝐹𝐶) → 𝐶 ∈ V) | |
4 | 2, 3 | sylan 582 | . 2 ⊢ ((Rel 𝐹 ∧ 〈𝐴, 𝐵〉tpos 𝐹𝐶) → 𝐶 ∈ V) |
5 | brrelex2 5606 | . 2 ⊢ ((Rel 𝐹 ∧ 〈𝐵, 𝐴〉𝐹𝐶) → 𝐶 ∈ V) | |
6 | brtpos 7901 | . 2 ⊢ (𝐶 ∈ V → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) | |
7 | 4, 5, 6 | pm5.21nd 800 | 1 ⊢ (Rel 𝐹 → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 Vcvv 3494 〈cop 4573 class class class wbr 5066 Rel wrel 5560 tpos ctpos 7891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 df-tpos 7892 |
This theorem is referenced by: (None) |
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