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Theorem brtposg 6254
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 5115 . . . . 5 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
21breq1d 4013 . . . 4 ((𝐴𝑉𝐵𝑊) → ( {⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
323adant3 1017 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ( {⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
43anbi2d 464 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶) ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴𝐹𝐶)))
5 brtpos2 6251 . . 3 (𝐶𝑋 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
653ad2ant3 1020 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
7 opexg 4228 . . . . . . . . 9 ((𝐵𝑊𝐴𝑉) → ⟨𝐵, 𝐴⟩ ∈ V)
87ancoms 268 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ⟨𝐵, 𝐴⟩ ∈ V)
98anim1i 340 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑋))
1093impa 1194 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑋))
11 breldmg 4833 . . . . . . 7 ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑋 ∧ ⟨𝐵, 𝐴𝐹𝐶) → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
12113expia 1205 . . . . . 6 ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑋) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
1310, 12syl 14 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
14 opelcnvg 4807 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
15143adant3 1017 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
1613, 15sylibrd 169 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
17 elun1 3302 . . . 4 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}))
1816, 17syl6 33 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅})))
1918pm4.71rd 394 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐴𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴𝐹𝐶)))
204, 6, 193bitr4d 220 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978  wcel 2148  Vcvv 2737  cun 3127  c0 3422  {csn 3592  cop 3595   cuni 3809   class class class wbr 4003  ccnv 4625  dom cdm 4626  tpos ctpos 6244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-tpos 6245
This theorem is referenced by:  ottposg  6255  dmtpos  6256  rntpos  6257  ovtposg  6259  dftpos3  6262  tpostpos  6264
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