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Theorem brtpos 7876
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 7873 . . . . 5 (𝐶𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
21adantr 484 . . . 4 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
3 opex 5329 . . . . . . . . . 10 𝐵, 𝐴⟩ ∈ V
4 breldmg 5751 . . . . . . . . . . 11 ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑉 ∧ ⟨𝐵, 𝐴𝐹𝐶) → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
543expia 1118 . . . . . . . . . 10 ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶𝑉) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
63, 5mpan 689 . . . . . . . . 9 (𝐶𝑉 → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
76adantr 484 . . . . . . . 8 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
8 opelcnvg 5724 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
98adantl 485 . . . . . . . 8 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
107, 9sylibrd 262 . . . . . . 7 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
11 elun1 4128 . . . . . . 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 566 . . . . 5 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴𝐹𝐶)))
14 opswap 6059 . . . . . . 7 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
1514breq1i 5046 . . . . . 6 ( {⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶)
1615anbi2i 625 . . . . 5 ((⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶) ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴𝐹𝐶))
1713, 16syl6bbr 292 . . . 4 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (dom 𝐹 ∪ {∅}) ∧ {⟨𝐴, 𝐵⟩}𝐹𝐶)))
182, 17bitr4d 285 . . 3 ((𝐶𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
1918ex 416 . 2 (𝐶𝑉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶)))
20 brtpos0 7874 . . 3 (𝐶𝑉 → (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶))
21 opprc 4799 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
2221breq1d 5049 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ∅tpos 𝐹𝐶))
23 ancom 464 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
24 opprc 4799 . . . . . 6 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
2524breq1d 5049 . . . . 5 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (⟨𝐵, 𝐴𝐹𝐶 ↔ ∅𝐹𝐶))
2623, 25sylnbi 333 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐵, 𝐴𝐹𝐶 ↔ ∅𝐹𝐶))
2722, 26bibi12d 349 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶) ↔ (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶)))
2820, 27syl5ibrcom 250 . 2 (𝐶𝑉 → (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶)))
2919, 28pm2.61d 182 1 (𝐶𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2115  Vcvv 3471  cun 3908  c0 4266  {csn 4540  cop 4546   cuni 4811   class class class wbr 5039  ccnv 5527  dom cdm 5528  tpos ctpos 7866
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-tpos 7867
This theorem is referenced by:  ottpos  7877  relbrtpos  7878  dmtpos  7879  rntpos  7880  ovtpos  7882  dftpos3  7885  tpostpos  7887
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