Theorem brtpos 8168

Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
brtpos	⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))

Proof of Theorem brtpos

Step	Hyp	Ref	Expression
1		brtpos2 8165	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
2	1	adantr 480	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
3		opex 5407	. . . . . . . . . 10 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4		breldmg 5852	. . . . . . . . . . 11 ⊢ ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ⟨𝐵, 𝐴⟩𝐹𝐶) → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
5	4	3expia 1121	. . . . . . . . . 10 ⊢ ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑉) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
6	3, 5	mpan 690	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
8		opelcnvg 5823	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
10	7, 9	sylibrd 259	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹))
11		elun1 4133	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 → ⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}))
12	10, 11	syl6 35	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅})))
13	12	pm4.71rd 562	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴⟩𝐹𝐶)))
14		opswap 6178	. . . . . . 7 ⊢ ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩
15	14	breq1i 5099	. . . . . 6 ⊢ (∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)
16	15	anbi2i 623	. . . . 5 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶) ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴⟩𝐹𝐶))
17	13, 16	bitr4di 289	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
18	2, 17	bitr4d 282	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))
19	18	ex 412	. 2 ⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)))
20		brtpos0 8166	. . 3 ⊢ (𝐶 ∈ 𝑉 → (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶))
21		opprc 4847	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
22	21	breq1d 5102	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ∅tpos 𝐹𝐶))
23		ancom 460	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
24		opprc 4847	. . . . . 6 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
25	24	breq1d 5102	. . . . 5 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ ∅𝐹𝐶))
26	23, 25	sylnbi 330	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ ∅𝐹𝐶))
27	22, 26	bibi12d 345	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶) ↔ (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶)))
28	20, 27	syl5ibrcom 247	. 2 ⊢ (𝐶 ∈ 𝑉 → (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)))
29	19, 28	pm2.61d 179	1 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3436   ∪ cun 3901  ∅c0 4284  {csn 4577  ⟨cop 4583  ∪ cuni 4858   class class class wbr 5092  ^◡ccnv 5618  dom cdm 5619  tpos ctpos 8158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490  df-tpos 8159

This theorem is referenced by:  ottpos  8169  relbrtpos  8170  dmtpos  8171  rntpos  8172  ovtpos  8174  dftpos3  8177  tpostpos  8179