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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 7357 | . . . 4 ⊢ Rel tpos 𝐹 | |
2 | 1 | a1i 11 | . . 3 ⊢ (Rel 𝐹 → Rel tpos 𝐹) |
3 | brrelex2 5157 | . . 3 ⊢ ((Rel tpos 𝐹 ∧ 〈𝐴, 𝐵〉tpos 𝐹𝐶) → 𝐶 ∈ V) | |
4 | 2, 3 | sylan 488 | . 2 ⊢ ((Rel 𝐹 ∧ 〈𝐴, 𝐵〉tpos 𝐹𝐶) → 𝐶 ∈ V) |
5 | brrelex2 5157 | . 2 ⊢ ((Rel 𝐹 ∧ 〈𝐵, 𝐴〉𝐹𝐶) → 𝐶 ∈ V) | |
6 | brtpos 7361 | . 2 ⊢ (𝐶 ∈ V → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) | |
7 | 4, 5, 6 | pm5.21nd 941 | 1 ⊢ (Rel 𝐹 → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 Vcvv 3200 〈cop 4183 class class class wbr 4653 Rel wrel 5119 tpos ctpos 7351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-tpos 7352 |
This theorem is referenced by: (None) |
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