Theorem brtposg 6419

Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.)

Assertion

Ref	Expression
brtposg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))

Proof of Theorem brtposg

Step	Hyp	Ref	Expression
1		opswapg 5223	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
2	1	breq1d 4098	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))
3	2	3adant3 1043	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))
4	3	anbi2d 464	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶) ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴⟩𝐹𝐶)))
5		brtpos2 6416	. . 3 ⊢ (𝐶 ∈ 𝑋 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
6	5	3ad2ant3 1046	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶)))
7		opexg 4320	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ⟨𝐵, 𝐴⟩ ∈ V)
8	7	ancoms 268	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐵, 𝐴⟩ ∈ V)
9	8	anim1i 340	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑋))
10	9	3impa 1220	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑋))
11		breldmg 4937	. . . . . . 7 ⊢ ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑋 ∧ ⟨𝐵, 𝐴⟩𝐹𝐶) → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
12	11	3expia 1231	. . . . . 6 ⊢ ((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
13	10, 12	syl 14	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
14		opelcnvg 4910	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
15	14	3adant3 1043	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹))
16	13, 15	sylibrd 169	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹))
17		elun1 3374	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 → ⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}))
18	16, 17	syl6 33	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅})))
19	18	pm4.71rd 394	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴⟩𝐹𝐶)))
20	4, 6, 19	3bitr4d 220	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   ∈ wcel 2202  Vcvv 2802   ∪ cun 3198  ∅c0 3494  {csn 3669  ⟨cop 3672  ∪ cuni 3893   class class class wbr 4088  ^◡ccnv 4724  dom cdm 4725  tpos ctpos 6409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-tpos 6410

This theorem is referenced by:  ottposg  6420  dmtpos  6421  rntpos  6422  ovtposg  6424  dftpos3  6427  tpostpos  6429