Theorem brtposg 5757

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 4701	. . . . 5 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∪ ◡{⟨A, B⟩} = ⟨B, A⟩)
2	1	breq1d 3726	. . . 4 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (∪ ◡{⟨A, B⟩}𝐹𝐶 ↔ ⟨B, A⟩𝐹𝐶))
3	2	3adant3 914	. . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∪ ◡{⟨A, B⟩}𝐹𝐶 ↔ ⟨B, A⟩𝐹𝐶))
4	3	anbi2d 440	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((⟨A, B⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨A, B⟩}𝐹𝐶) ↔ (⟨A, B⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨B, A⟩𝐹𝐶)))
5		brtpos2 5754	. . 3 ⊢ (𝐶 ∈ 𝑋 → (⟨A, B⟩tpos 𝐹𝐶 ↔ (⟨A, B⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨A, B⟩}𝐹𝐶)))
6	5	3ad2ant3 917	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨A, B⟩tpos 𝐹𝐶 ↔ (⟨A, B⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨A, B⟩}𝐹𝐶)))
7		opexg 3916	. . . . . . . . 9 ⊢ ((B ∈ 𝑊 ∧ A ∈ 𝑉) → ⟨B, A⟩ ∈ V)
8	7	ancoms 255	. . . . . . . 8 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ⟨B, A⟩ ∈ V)
9	8	anim1i 323	. . . . . . 7 ⊢ (((A ∈ 𝑉 ∧ B ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (⟨B, A⟩ ∈ V ∧ 𝐶 ∈ 𝑋))
10	9	3impa 1087	. . . . . 6 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨B, A⟩ ∈ V ∧ 𝐶 ∈ 𝑋))
11		breldmg 4434	. . . . . . 7 ⊢ ((⟨B, A⟩ ∈ V ∧ 𝐶 ∈ 𝑋 ∧ ⟨B, A⟩𝐹𝐶) → ⟨B, A⟩ ∈ dom 𝐹)
12	11	3expia 1094	. . . . . 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 4408	. . . . . 6 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (⟨A, B⟩ ∈ ◡dom 𝐹 ↔ ⟨B, A⟩ ∈ dom 𝐹))
15	14	3adant3 914	. . . . 5 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨A, B⟩ ∈ ◡dom 𝐹 ↔ ⟨B, A⟩ ∈ dom 𝐹))
16	13, 15	sylibrd 158	. . . 4 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨B, A⟩𝐹𝐶 → ⟨A, B⟩ ∈ ◡dom 𝐹))
17		elun1 3088	. . . 4 ⊢ (⟨A, B⟩ ∈ ◡dom 𝐹 → ⟨A, B⟩ ∈ (◡dom 𝐹 ∪ {∅}))
18	16, 17	syl6 29	. . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨B, A⟩𝐹𝐶 → ⟨A, B⟩ ∈ (◡dom 𝐹 ∪ {∅})))
19	18	pm4.71rd 374	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨B, A⟩𝐹𝐶 ↔ (⟨A, B⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨B, A⟩𝐹𝐶)))
20	4, 6, 19	3bitr4d 209	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨A, B⟩tpos 𝐹𝐶 ↔ ⟨B, A⟩𝐹𝐶))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 875   ∈ wcel 1375  Vcvv 2533   ∪ cun 2893  ∅c0 3202  {csn 3327  ⟨cop 3330  ∪ cuni 3532   class class class wbr 3716  ^◡ccnv 4237  dom cdm 4238  tpos ctpos 5747

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093

This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-rab 2291  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-mpt 3772  df-id 3983  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-res 4250  df-ima 4251  df-iota 4761  df-fun 4798  df-fn 4799  df-fv 4804  df-tpos 5748

This theorem is referenced by:  ottposg  5758  dmtpos  5759  rntpos  5760  ovtposg  5762  dftpos3  5765  tpostpos  5767