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Theorem brtpos 8222
Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
brtpos (𝐶𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))

Proof of Theorem brtpos
StepHypRef Expression
1 brtpos2 8219 . . . . 5 (𝐶𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
21adantr 481 . . . 4 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
3 opex 5464 . . . . . . . . . 10 𝐵, 𝐴⟩ ∈ V
4 breldmg 5909 . . . . . . . . . . 11 ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑉 ∧ ⟨𝐵, 𝐴𝐹𝐶) → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
543expia 1121 . . . . . . . . . 10 ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑉) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
63, 5mpan 688 . . . . . . . . 9 (𝐶𝑉 → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
76adantr 481 . . . . . . . 8 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
8 opelcnvg 5880 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
98adantl 482 . . . . . . . 8 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
107, 9sylibrd 258 . . . . . . 7 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
11 elun1 4176 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}))
1210, 11syl6 35 . . . . . 6 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅})))
1312pm4.71rd 563 . . . . 5 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴𝐹𝐶)))
14 opswap 6228 . . . . . . 7 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
1514breq1i 5155 . . . . . 6 ( {⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶)
1615anbi2i 623 . . . . 5 ((⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶) ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴𝐹𝐶))
1713, 16bitr4di 288 . . . 4 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
182, 17bitr4d 281 . . 3 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
1918ex 413 . 2 (𝐶𝑉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶)))
20 brtpos0 8220 . . 3 (𝐶𝑉 → (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶))
21 opprc 4896 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
2221breq1d 5158 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ∅tpos 𝐹𝐶))
23 ancom 461 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
24 opprc 4896 . . . . . 6 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
2524breq1d 5158 . . . . 5 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (⟨𝐵, 𝐴𝐹𝐶 ↔ ∅𝐹𝐶))
2623, 25sylnbi 329 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐵, 𝐴𝐹𝐶 ↔ ∅𝐹𝐶))
2722, 26bibi12d 345 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶) ↔ (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶)))
2820, 27syl5ibrcom 246 . 2 (𝐶𝑉 → (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶)))
2919, 28pm2.61d 179 1 (𝐶𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wcel 2106  Vcvv 3474  cun 3946  c0 4322  {csn 4628  cop 4634   cuni 4908   class class class wbr 5148  ccnv 5675  dom cdm 5676  tpos ctpos 8212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-tpos 8213
This theorem is referenced by:  ottpos  8223  relbrtpos  8224  dmtpos  8225  rntpos  8226  ovtpos  8228  dftpos3  8231  tpostpos  8233
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