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Theorem tpostpos 6011
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 5997 . 2 Rel tpos tpos 𝐹
2 inss2 3219 . . 3 (𝐹 ∩ (((V × V) ∪ {∅}) × V)) ⊆ (((V × V) ∪ {∅}) × V)
3 relxp 4535 . . 3 Rel (((V × V) ∪ {∅}) × V)
4 relss 4513 . . 3 ((𝐹 ∩ (((V × V) ∪ {∅}) × V)) ⊆ (((V × V) ∪ {∅}) × V) → (Rel (((V × V) ∪ {∅}) × V) → Rel (𝐹 ∩ (((V × V) ∪ {∅}) × V))))
52, 3, 4mp2 16 . 2 Rel (𝐹 ∩ (((V × V) ∪ {∅}) × V))
6 relcnv 4797 . . . . . . . . 9 Rel dom tpos 𝐹
7 df-rel 4435 . . . . . . . . 9 (Rel dom tpos 𝐹dom tpos 𝐹 ⊆ (V × V))
86, 7mpbi 143 . . . . . . . 8 dom tpos 𝐹 ⊆ (V × V)
9 simpl 107 . . . . . . . 8 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) → 𝑤dom tpos 𝐹)
108, 9sseldi 3021 . . . . . . 7 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) → 𝑤 ∈ (V × V))
11 simpr 108 . . . . . . 7 ((𝑤𝐹𝑧𝑤 ∈ (V × V)) → 𝑤 ∈ (V × V))
12 elvv 4488 . . . . . . . . 9 (𝑤 ∈ (V × V) ↔ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
13 eleq1 2150 . . . . . . . . . . . . . 14 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹))
14 vex 2622 . . . . . . . . . . . . . . 15 𝑥 ∈ V
15 vex 2622 . . . . . . . . . . . . . . 15 𝑦 ∈ V
1614, 15opelcnv 4606 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹)
1713, 16syl6bb 194 . . . . . . . . . . . . 13 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤dom tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹))
18 sneq 3452 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
1918cnveqd 4600 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
2019unieqd 3659 . . . . . . . . . . . . . . 15 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
21 opswapg 4904 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → {⟨𝑥, 𝑦⟩} = ⟨𝑦, 𝑥⟩)
2214, 15, 21mp2an 417 . . . . . . . . . . . . . . 15 {⟨𝑥, 𝑦⟩} = ⟨𝑦, 𝑥
2320, 22syl6eq 2136 . . . . . . . . . . . . . 14 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = ⟨𝑦, 𝑥⟩)
2423breq1d 3847 . . . . . . . . . . . . 13 (𝑤 = ⟨𝑥, 𝑦⟩ → ( {𝑤}tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
2517, 24anbi12d 457 . . . . . . . . . . . 12 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧)))
2615, 14opex 4047 . . . . . . . . . . . . . . 15 𝑦, 𝑥⟩ ∈ V
27 vex 2622 . . . . . . . . . . . . . . 15 𝑧 ∈ V
2826, 27breldm 4628 . . . . . . . . . . . . . 14 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 → ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹)
2928pm4.71ri 384 . . . . . . . . . . . . 13 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
30 brtposg 6001 . . . . . . . . . . . . . 14 ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧))
3115, 14, 27, 30mp3an 1273 . . . . . . . . . . . . 13 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧)
3229, 31bitr3i 184 . . . . . . . . . . . 12 ((⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧) ↔ ⟨𝑥, 𝑦𝐹𝑧)
3325, 32syl6bb 194 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ ⟨𝑥, 𝑦𝐹𝑧))
34 breq1 3840 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧))
3533, 34bitr4d 189 . . . . . . . . . 10 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
3635exlimivv 1824 . . . . . . . . 9 (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
3712, 36sylbi 119 . . . . . . . 8 (𝑤 ∈ (V × V) → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
38 iba 294 . . . . . . . 8 (𝑤 ∈ (V × V) → (𝑤𝐹𝑧 ↔ (𝑤𝐹𝑧𝑤 ∈ (V × V))))
3937, 38bitrd 186 . . . . . . 7 (𝑤 ∈ (V × V) → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ (V × V))))
4010, 11, 39pm5.21nii 655 . . . . . 6 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ (V × V)))
41 elsni 3459 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {∅} → 𝑤 = ∅)
4241sneqd 3454 . . . . . . . . . . . . . . 15 (𝑤 ∈ {∅} → {𝑤} = {∅})
4342cnveqd 4600 . . . . . . . . . . . . . 14 (𝑤 ∈ {∅} → {𝑤} = {∅})
44 cnvsn0 4886 . . . . . . . . . . . . . 14 {∅} = ∅
4543, 44syl6eq 2136 . . . . . . . . . . . . 13 (𝑤 ∈ {∅} → {𝑤} = ∅)
4645unieqd 3659 . . . . . . . . . . . 12 (𝑤 ∈ {∅} → {𝑤} = ∅)
47 uni0 3675 . . . . . . . . . . . 12 ∅ = ∅
4846, 47syl6eq 2136 . . . . . . . . . . 11 (𝑤 ∈ {∅} → {𝑤} = ∅)
4948breq1d 3847 . . . . . . . . . 10 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧 ↔ ∅tpos 𝐹𝑧))
50 brtpos0 5999 . . . . . . . . . . 11 (𝑧 ∈ V → (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
5127, 50ax-mp 7 . . . . . . . . . 10 (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧)
5249, 51syl6bb 194 . . . . . . . . 9 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
5341breq1d 3847 . . . . . . . . 9 (𝑤 ∈ {∅} → (𝑤𝐹𝑧 ↔ ∅𝐹𝑧))
5452, 53bitr4d 189 . . . . . . . 8 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧𝑤𝐹𝑧))
5554pm5.32i 442 . . . . . . 7 ((𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧))
56 ancom 262 . . . . . . 7 ((𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ {∅}))
5755, 56bitri 182 . . . . . 6 ((𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ {∅}))
5840, 57orbi12i 716 . . . . 5 (((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧)) ↔ ((𝑤𝐹𝑧𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧𝑤 ∈ {∅})))
59 andir 768 . . . . 5 (((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧)))
60 andi 767 . . . . 5 ((𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})) ↔ ((𝑤𝐹𝑧𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧𝑤 ∈ {∅})))
6158, 59, 603bitr4i 210 . . . 4 (((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})))
62 elun 3139 . . . . 5 (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ↔ (𝑤dom tpos 𝐹𝑤 ∈ {∅}))
6362anbi1i 446 . . . 4 ((𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ ((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧))
64 brxp 4458 . . . . . . 7 (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 ∈ V))
6527, 64mpbiran2 887 . . . . . 6 (𝑤(((V × V) ∪ {∅}) × V)𝑧𝑤 ∈ ((V × V) ∪ {∅}))
66 elun 3139 . . . . . 6 (𝑤 ∈ ((V × V) ∪ {∅}) ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))
6765, 66bitri 182 . . . . 5 (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))
6867anbi2i 445 . . . 4 ((𝑤𝐹𝑧𝑤(((V × V) ∪ {∅}) × V)𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})))
6961, 63, 683bitr4i 210 . . 3 ((𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤(((V × V) ∪ {∅}) × V)𝑧))
70 brtpos2 5998 . . . 4 (𝑧 ∈ V → (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧)))
7127, 70ax-mp 7 . . 3 (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧))
72 brin 3884 . . 3 (𝑤(𝐹 ∩ (((V × V) ∪ {∅}) × V))𝑧 ↔ (𝑤𝐹𝑧𝑤(((V × V) ∪ {∅}) × V)𝑧))
7369, 71, 723bitr4i 210 . 2 (𝑤tpos tpos 𝐹𝑧𝑤(𝐹 ∩ (((V × V) ∪ {∅}) × V))𝑧)
741, 5, 73eqbrriv 4521 1 tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wo 664   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  cun 2995  cin 2996  wss 2997  c0 3284  {csn 3441  cop 3444   cuni 3648   class class class wbr 3837   × cxp 4426  ccnv 4427  dom cdm 4428  Rel wrel 4433  tpos ctpos 5991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010  df-tpos 5992
This theorem is referenced by:  tpostpos2  6012
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