Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  brtposg GIF version

Theorem brtposg 5869
 Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.)
Assertion
Ref Expression
brtposg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))

Proof of Theorem brtposg
StepHypRef Expression
1 opswapg 4807 . . . . 5 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
21breq1d 3774 . . . 4 ((𝐴𝑉𝐵𝑊) → ( {⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
323adant3 924 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ( {⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
43anbi2d 437 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶) ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴𝐹𝐶)))
5 brtpos2 5866 . . 3 (𝐶𝑋 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
653ad2ant3 927 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
7 opexg 3964 . . . . . . . . 9 ((𝐵𝑊𝐴𝑉) → ⟨𝐵, 𝐴⟩ ∈ V)
87ancoms 255 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ⟨𝐵, 𝐴⟩ ∈ V)
98anim1i 323 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑋))
1093impa 1099 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑋))
11 breldmg 4541 . . . . . . 7 ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑋 ∧ ⟨𝐵, 𝐴𝐹𝐶) → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
12113expia 1106 . . . . . 6 ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑋) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
1310, 12syl 14 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
14 opelcnvg 4515 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
15143adant3 924 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
1613, 15sylibrd 158 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
17 elun1 3110 . . . 4 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}))
1816, 17syl6 29 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅})))
1918pm4.71rd 374 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐴𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴𝐹𝐶)))
204, 6, 193bitr4d 209 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   ∈ wcel 1393  Vcvv 2557   ∪ cun 2915  ∅c0 3224  {csn 3375  ⟨cop 3378  ∪ cuni 3580   class class class wbr 3764  ◡ccnv 4344  dom cdm 4345  tpos ctpos 5859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-tpos 5860 This theorem is referenced by:  ottposg  5870  dmtpos  5871  rntpos  5872  ovtposg  5874  dftpos3  5877  tpostpos  5879
 Copyright terms: Public domain W3C validator