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Theorem tpostpos 8272
Description: Value of the double transposition for a general class 𝐹. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
tpostpos tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))

Proof of Theorem tpostpos
Dummy variables 𝑥 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reltpos 8257 . 2 Rel tpos tpos 𝐹
2 relinxp 5823 . 2 Rel (𝐹 ∩ (((V × V) ∪ {∅}) × V))
3 relcnv 6121 . . . . . . . . 9 Rel dom tpos 𝐹
4 df-rel 5691 . . . . . . . . 9 (Rel dom tpos 𝐹dom tpos 𝐹 ⊆ (V × V))
53, 4mpbi 230 . . . . . . . 8 dom tpos 𝐹 ⊆ (V × V)
6 simpl 482 . . . . . . . 8 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) → 𝑤dom tpos 𝐹)
75, 6sselid 3980 . . . . . . 7 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) → 𝑤 ∈ (V × V))
8 simpr 484 . . . . . . 7 ((𝑤𝐹𝑧𝑤 ∈ (V × V)) → 𝑤 ∈ (V × V))
9 elvv 5759 . . . . . . . . 9 (𝑤 ∈ (V × V) ↔ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
10 eleq1 2828 . . . . . . . . . . . . . 14 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹))
11 vex 3483 . . . . . . . . . . . . . . 15 𝑥 ∈ V
12 vex 3483 . . . . . . . . . . . . . . 15 𝑦 ∈ V
1311, 12opelcnv 5891 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹)
1410, 13bitrdi 287 . . . . . . . . . . . . 13 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤dom tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹))
15 sneq 4635 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
1615cnveqd 5885 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
1716unieqd 4919 . . . . . . . . . . . . . . 15 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
18 opswap 6248 . . . . . . . . . . . . . . 15 {⟨𝑥, 𝑦⟩} = ⟨𝑦, 𝑥
1917, 18eqtrdi 2792 . . . . . . . . . . . . . 14 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = ⟨𝑦, 𝑥⟩)
2019breq1d 5152 . . . . . . . . . . . . 13 (𝑤 = ⟨𝑥, 𝑦⟩ → ( {𝑤}tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
2114, 20anbi12d 632 . . . . . . . . . . . 12 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧)))
22 opex 5468 . . . . . . . . . . . . . . 15 𝑦, 𝑥⟩ ∈ V
23 vex 3483 . . . . . . . . . . . . . . 15 𝑧 ∈ V
2422, 23breldm 5918 . . . . . . . . . . . . . 14 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 → ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹)
2524pm4.71ri 560 . . . . . . . . . . . . 13 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
26 brtpos 8261 . . . . . . . . . . . . . 14 (𝑧 ∈ V → (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧))
2726elv 3484 . . . . . . . . . . . . 13 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧)
2825, 27bitr3i 277 . . . . . . . . . . . 12 ((⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧) ↔ ⟨𝑥, 𝑦𝐹𝑧)
2921, 28bitrdi 287 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ ⟨𝑥, 𝑦𝐹𝑧))
30 breq1 5145 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧))
3129, 30bitr4d 282 . . . . . . . . . 10 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
3231exlimivv 1931 . . . . . . . . 9 (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
339, 32sylbi 217 . . . . . . . 8 (𝑤 ∈ (V × V) → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
34 iba 527 . . . . . . . 8 (𝑤 ∈ (V × V) → (𝑤𝐹𝑧 ↔ (𝑤𝐹𝑧𝑤 ∈ (V × V))))
3533, 34bitrd 279 . . . . . . 7 (𝑤 ∈ (V × V) → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ (V × V))))
367, 8, 35pm5.21nii 378 . . . . . 6 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ (V × V)))
37 elsni 4642 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {∅} → 𝑤 = ∅)
3837sneqd 4637 . . . . . . . . . . . . . . 15 (𝑤 ∈ {∅} → {𝑤} = {∅})
3938cnveqd 5885 . . . . . . . . . . . . . 14 (𝑤 ∈ {∅} → {𝑤} = {∅})
40 cnvsn0 6229 . . . . . . . . . . . . . 14 {∅} = ∅
4139, 40eqtrdi 2792 . . . . . . . . . . . . 13 (𝑤 ∈ {∅} → {𝑤} = ∅)
4241unieqd 4919 . . . . . . . . . . . 12 (𝑤 ∈ {∅} → {𝑤} = ∅)
43 uni0 4934 . . . . . . . . . . . 12 ∅ = ∅
4442, 43eqtrdi 2792 . . . . . . . . . . 11 (𝑤 ∈ {∅} → {𝑤} = ∅)
4544breq1d 5152 . . . . . . . . . 10 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧 ↔ ∅tpos 𝐹𝑧))
46 brtpos0 8259 . . . . . . . . . . 11 (𝑧 ∈ V → (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
4746elv 3484 . . . . . . . . . 10 (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧)
4845, 47bitrdi 287 . . . . . . . . 9 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
4937breq1d 5152 . . . . . . . . 9 (𝑤 ∈ {∅} → (𝑤𝐹𝑧 ↔ ∅𝐹𝑧))
5048, 49bitr4d 282 . . . . . . . 8 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧𝑤𝐹𝑧))
5150pm5.32i 574 . . . . . . 7 ((𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧))
5251biancomi 462 . . . . . 6 ((𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ {∅}))
5336, 52orbi12i 914 . . . . 5 (((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧)) ↔ ((𝑤𝐹𝑧𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧𝑤 ∈ {∅})))
54 andir 1010 . . . . 5 (((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧)))
55 andi 1009 . . . . 5 ((𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})) ↔ ((𝑤𝐹𝑧𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧𝑤 ∈ {∅})))
5653, 54, 553bitr4i 303 . . . 4 (((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})))
57 elun 4152 . . . . 5 (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ↔ (𝑤dom tpos 𝐹𝑤 ∈ {∅}))
5857anbi1i 624 . . . 4 ((𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ ((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧))
59 brxp 5733 . . . . . . 7 (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 ∈ V))
6023, 59mpbiran2 710 . . . . . 6 (𝑤(((V × V) ∪ {∅}) × V)𝑧𝑤 ∈ ((V × V) ∪ {∅}))
61 elun 4152 . . . . . 6 (𝑤 ∈ ((V × V) ∪ {∅}) ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))
6260, 61bitri 275 . . . . 5 (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))
6362anbi2i 623 . . . 4 ((𝑤𝐹𝑧𝑤(((V × V) ∪ {∅}) × V)𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})))
6456, 58, 633bitr4i 303 . . 3 ((𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤(((V × V) ∪ {∅}) × V)𝑧))
65 brtpos2 8258 . . . 4 (𝑧 ∈ V → (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧)))
6665elv 3484 . . 3 (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧))
67 brin 5194 . . 3 (𝑤(𝐹 ∩ (((V × V) ∪ {∅}) × V))𝑧 ↔ (𝑤𝐹𝑧𝑤(((V × V) ∪ {∅}) × V)𝑧))
6864, 66, 673bitr4i 303 . 2 (𝑤tpos tpos 𝐹𝑧𝑤(𝐹 ∩ (((V × V) ∪ {∅}) × V))𝑧)
691, 2, 68eqbrriv 5800 1 tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wo 847   = wceq 1539  wex 1778  wcel 2107  Vcvv 3479  cun 3948  cin 3949  wss 3950  c0 4332  {csn 4625  cop 4631   cuni 4906   class class class wbr 5142   × cxp 5682  ccnv 5683  dom cdm 5684  Rel wrel 5689  tpos ctpos 8251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-tpos 8252
This theorem is referenced by:  tpostpos2  8273
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