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Theorem brtpos 7248
 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 7245 . . . . 5 (𝐶𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
21adantr 480 . . . 4 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
3 opex 4859 . . . . . . . . . 10 𝐵, 𝐴⟩ ∈ V
4 breldmg 5252 . . . . . . . . . . 11 ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑉 ∧ ⟨𝐵, 𝐴𝐹𝐶) → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
543expia 1259 . . . . . . . . . 10 ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑉) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
63, 5mpan 702 . . . . . . . . 9 (𝐶𝑉 → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
76adantr 480 . . . . . . . 8 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
8 opelcnvg 5224 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
98adantl 481 . . . . . . . 8 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
107, 9sylibrd 248 . . . . . . 7 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
11 elun1 3742 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}))
1210, 11syl6 34 . . . . . 6 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅})))
1312pm4.71rd 665 . . . . 5 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴𝐹𝐶)))
14 opswap 5540 . . . . . . 7 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
1514breq1i 4590 . . . . . 6 ( {⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶)
1615anbi2i 726 . . . . 5 ((⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶) ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴𝐹𝐶))
1713, 16syl6bbr 277 . . . 4 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
182, 17bitr4d 270 . . 3 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
1918ex 449 . 2 (𝐶𝑉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶)))
20 brtpos0 7246 . . 3 (𝐶𝑉 → (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶))
21 opprc 4362 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
2221breq1d 4593 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ∅tpos 𝐹𝐶))
23 ancom 465 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
24 opprc 4362 . . . . . 6 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
2524breq1d 4593 . . . . 5 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (⟨𝐵, 𝐴𝐹𝐶 ↔ ∅𝐹𝐶))
2623, 25sylnbi 319 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐵, 𝐴𝐹𝐶 ↔ ∅𝐹𝐶))
2722, 26bibi12d 334 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶) ↔ (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶)))
2820, 27syl5ibrcom 236 . 2 (𝐶𝑉 → (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶)))
2919, 28pm2.61d 169 1 (𝐶𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  tpos ctpos 7238 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-tpos 7239 This theorem is referenced by:  ottpos  7249  relbrtpos  7250  dmtpos  7251  rntpos  7252  ovtpos  7254  dftpos3  7257  tpostpos  7259
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