MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relbrtpos Structured version   Visualization version   GIF version

Theorem relbrtpos 7514
Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
relbrtpos (Rel 𝐹 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))

Proof of Theorem relbrtpos
StepHypRef Expression
1 reltpos 7508 . . . 4 Rel tpos 𝐹
21a1i 11 . . 3 (Rel 𝐹 → Rel tpos 𝐹)
3 brrelex2 5297 . . 3 ((Rel tpos 𝐹 ∧ ⟨𝐴, 𝐵⟩tpos 𝐹𝐶) → 𝐶 ∈ V)
42, 3sylan 561 . 2 ((Rel 𝐹 ∧ ⟨𝐴, 𝐵⟩tpos 𝐹𝐶) → 𝐶 ∈ V)
5 brrelex2 5297 . 2 ((Rel 𝐹 ∧ ⟨𝐵, 𝐴𝐹𝐶) → 𝐶 ∈ V)
6 brtpos 7512 . 2 (𝐶 ∈ V → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
74, 5, 6pm5.21nd 803 1 (Rel 𝐹 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2145  Vcvv 3351  cop 4322   class class class wbr 4786  Rel wrel 5254  tpos ctpos 7502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-tpos 7503
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator