Theorem brtpos 8263

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 8260	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
2	1	adantr 486	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
3		opex 5435	. . . . . . . . . 10 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4		breldmg 5895	. . . . . . . . . . 11 ⊢ ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ⟨𝐵, 𝐴⟩𝐹𝐶) → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
5	4	3expia 1139	. . . . . . . . . 10 ⊢ ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑉) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
6	3, 5	mpan 703	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
7	6	adantr 486	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
8		opelcnvg 5862	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
9	8	adantl 487	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
10	7, 9	sylibrd 262	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹))
11		elun1 4182	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 → ⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}))
12	10, 11	syl6 36	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅})))
13	12	pm4.71rd 572	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴⟩𝐹𝐶)))
14		opswap 6236	. . . . . . 7 ⊢ ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩
15	14	breq1i 5112	. . . . . 6 ⊢ (∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)
16	15	anbi2i 635	. . . . 5 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶) ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴⟩𝐹𝐶))
17	13, 16	bitr4di 292	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
18	2, 17	bitr4d 285	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))
19	18	ex 418	. 2 ⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)))
20		brtpos0 8261	. . 3 ⊢ (𝐶 ∈ 𝑉 → (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶))
21		opprc 4856	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
22	21	breq1d 5115	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ∅tpos 𝐹𝐶))
23		ancom 466	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
24		opprc 4856	. . . . . 6 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
25	24	breq1d 5115	. . . . 5 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ ∅𝐹𝐶))
26	23, 25	sylnbi 333	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ ∅𝐹𝐶))
27	22, 26	bibi12d 348	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶) ↔ (∅tpos 𝐹𝐶 ↔ ∅𝐹𝐶)))
28	20, 27	syl5ibrcom 250	. 2 ⊢ (𝐶 ∈ 𝑉 → (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)))
29	19, 28	pm2.61d 181	1 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  Vcvv 3453   ∪ cun 4151  ∅c0 4280  {csn 4587  ⟨cop 4837  ∪ cuni 4869   class class class wbr 5105  ^◡ccnv 5653  dom cdm 5654  tpos ctpos 8253

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-fv 6554  df-tpos 8254

This theorem is used by:  ottpos  8264  relbrtpos  8265  dmtpos  8266  rntpos  8267  ovtpos  8269  dftpos3  8272  tpostpos  8274