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Theorem tpostpos 8181
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 8166 . 2 Rel tpos tpos 𝐹
2 relinxp 5774 . 2 Rel (𝐹 ∩ (((V × V) ∪ {∅}) × V))
3 relcnv 6060 . . . . . . . . 9 Rel dom tpos 𝐹
4 df-rel 5644 . . . . . . . . 9 (Rel dom tpos 𝐹dom tpos 𝐹 ⊆ (V × V))
53, 4mpbi 229 . . . . . . . 8 dom tpos 𝐹 ⊆ (V × V)
6 simpl 484 . . . . . . . 8 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) → 𝑤dom tpos 𝐹)
75, 6sselid 3946 . . . . . . 7 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) → 𝑤 ∈ (V × V))
8 simpr 486 . . . . . . 7 ((𝑤𝐹𝑧𝑤 ∈ (V × V)) → 𝑤 ∈ (V × V))
9 elvv 5710 . . . . . . . . 9 (𝑤 ∈ (V × V) ↔ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
10 eleq1 2822 . . . . . . . . . . . . . 14 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹))
11 vex 3451 . . . . . . . . . . . . . . 15 𝑥 ∈ V
12 vex 3451 . . . . . . . . . . . . . . 15 𝑦 ∈ V
1311, 12opelcnv 5841 . . . . . . . . . . . . . 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 4600 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
1615cnveqd 5835 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
1716unieqd 4883 . . . . . . . . . . . . . . 15 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
18 opswap 6185 . . . . . . . . . . . . . . 15 {⟨𝑥, 𝑦⟩} = ⟨𝑦, 𝑥
1917, 18eqtrdi 2789 . . . . . . . . . . . . . 14 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = ⟨𝑦, 𝑥⟩)
2019breq1d 5119 . . . . . . . . . . . . 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 5425 . . . . . . . . . . . . . . 15 𝑦, 𝑥⟩ ∈ V
23 vex 3451 . . . . . . . . . . . . . . 15 𝑧 ∈ V
2422, 23breldm 5868 . . . . . . . . . . . . . 14 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 → ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹)
2524pm4.71ri 562 . . . . . . . . . . . . 13 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
26 brtpos 8170 . . . . . . . . . . . . . 14 (𝑧 ∈ V → (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧))
2726elv 3453 . . . . . . . . . . . . 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 5112 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧))
3129, 30bitr4d 282 . . . . . . . . . 10 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
3231exlimivv 1936 . . . . . . . . 9 (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
339, 32sylbi 216 . . . . . . . 8 (𝑤 ∈ (V × V) → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
34 iba 529 . . . . . . . 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 380 . . . . . 6 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ (V × V)))
37 elsni 4607 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {∅} → 𝑤 = ∅)
3837sneqd 4602 . . . . . . . . . . . . . . 15 (𝑤 ∈ {∅} → {𝑤} = {∅})
3938cnveqd 5835 . . . . . . . . . . . . . 14 (𝑤 ∈ {∅} → {𝑤} = {∅})
40 cnvsn0 6166 . . . . . . . . . . . . . 14 {∅} = ∅
4139, 40eqtrdi 2789 . . . . . . . . . . . . 13 (𝑤 ∈ {∅} → {𝑤} = ∅)
4241unieqd 4883 . . . . . . . . . . . 12 (𝑤 ∈ {∅} → {𝑤} = ∅)
43 uni0 4900 . . . . . . . . . . . 12 ∅ = ∅
4442, 43eqtrdi 2789 . . . . . . . . . . 11 (𝑤 ∈ {∅} → {𝑤} = ∅)
4544breq1d 5119 . . . . . . . . . 10 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧 ↔ ∅tpos 𝐹𝑧))
46 brtpos0 8168 . . . . . . . . . . 11 (𝑧 ∈ V → (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
4746elv 3453 . . . . . . . . . 10 (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧)
4845, 47bitrdi 287 . . . . . . . . 9 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
4937breq1d 5119 . . . . . . . . 9 (𝑤 ∈ {∅} → (𝑤𝐹𝑧 ↔ ∅𝐹𝑧))
5048, 49bitr4d 282 . . . . . . . 8 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧𝑤𝐹𝑧))
5150pm5.32i 576 . . . . . . 7 ((𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧))
5251biancomi 464 . . . . . 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 1008 . . . . 5 (((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧)))
55 andi 1007 . . . . 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 4112 . . . . 5 (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ↔ (𝑤dom tpos 𝐹𝑤 ∈ {∅}))
5857anbi1i 625 . . . 4 ((𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ ((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧))
59 brxp 5685 . . . . . . 7 (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 ∈ V))
6023, 59mpbiran2 709 . . . . . 6 (𝑤(((V × V) ∪ {∅}) × V)𝑧𝑤 ∈ ((V × V) ∪ {∅}))
61 elun 4112 . . . . . 6 (𝑤 ∈ ((V × V) ∪ {∅}) ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))
6260, 61bitri 275 . . . . 5 (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))
6362anbi2i 624 . . . 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 8167 . . . 4 (𝑧 ∈ V → (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧)))
6665elv 3453 . . 3 (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧))
67 brin 5161 . . 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 5751 1 tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  Vcvv 3447  cun 3912  cin 3913  wss 3914  c0 4286  {csn 4590  cop 4596   cuni 4869   class class class wbr 5109   × cxp 5635  ccnv 5636  dom cdm 5637  Rel wrel 5642  tpos ctpos 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-tpos 8161
This theorem is referenced by:  tpostpos2  8182
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