Theorem brtpos 8175

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 8172	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
2	1	adantr 481	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
3		opex 5403	. . . . . . . . . 10 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4		breldmg 5851	. . . . . . . . . . 11 ⊢ ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ⟨𝐵, 𝐴⟩𝐹𝐶) → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
5	4	3expia 1127	. . . . . . . . . 10 ⊢ ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑉) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
6	3, 5	mpan 696	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
7	6	adantr 481	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
8		opelcnvg 5822	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
9	8	adantl 482	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
10	7, 9	sylibrd 260	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹))
11		elun1 4111	. . . . . . 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 567	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴⟩𝐹𝐶)))
14		opswap 6180	. . . . . . 7 ⊢ ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩
15	14	breq1i 5079	. . . . . 6 ⊢ (∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)
16	15	anbi2i 629	. . . . 5 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶) ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴⟩𝐹𝐶))
17	13, 16	bitr4di 290	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
18	2, 17	bitr4d 283	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))
19	18	ex 413	. 2 ⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)))
20		brtpos0 8173	. . 3 ⊢ (𝐶 ∈ 𝑉 → (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶))
21		opprc 4827	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
22	21	breq1d 5082	. . . 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 4827	. . . . . 6 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
25	24	breq1d 5082	. . . . 5 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ ∅𝐹𝐶))
26	23, 25	sylnbi 331	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ ∅𝐹𝐶))
27	22, 26	bibi12d 346	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶) ↔ (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶)))
28	20, 27	syl5ibrcom 248	. 2 ⊢ (𝐶 ∈ 𝑉 → (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)))
29	19, 28	pm2.61d 180	1 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881  ∅c0 4261  {csn 4555  ⟨cop 4561  ∪ cuni 4838   class class class wbr 5072  ^◡ccnv 5617  dom cdm 5618  tpos ctpos 8165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-fv 6493  df-tpos 8166

This theorem is referenced by:  ottpos  8176  relbrtpos  8177  dmtpos  8178  rntpos  8179  ovtpos  8181  dftpos3  8184  tpostpos  8186