MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpostpos Structured version   Visualization version   GIF version

Theorem tpostpos 7575
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 7560 . 2 Rel tpos tpos 𝐹
2 inss2 3993 . . 3 (𝐹 ∩ (((V × V) ∪ {∅}) × V)) ⊆ (((V × V) ∪ {∅}) × V)
3 relxp 5295 . . 3 Rel (((V × V) ∪ {∅}) × V)
4 relss 5376 . . 3 ((𝐹 ∩ (((V × V) ∪ {∅}) × V)) ⊆ (((V × V) ∪ {∅}) × V) → (Rel (((V × V) ∪ {∅}) × V) → Rel (𝐹 ∩ (((V × V) ∪ {∅}) × V))))
52, 3, 4mp2 9 . 2 Rel (𝐹 ∩ (((V × V) ∪ {∅}) × V))
6 relcnv 5685 . . . . . . . . 9 Rel dom tpos 𝐹
7 df-rel 5284 . . . . . . . . 9 (Rel dom tpos 𝐹dom tpos 𝐹 ⊆ (V × V))
86, 7mpbi 221 . . . . . . . 8 dom tpos 𝐹 ⊆ (V × V)
9 simpl 474 . . . . . . . 8 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) → 𝑤dom tpos 𝐹)
108, 9sseldi 3759 . . . . . . 7 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) → 𝑤 ∈ (V × V))
11 simpr 477 . . . . . . 7 ((𝑤𝐹𝑧𝑤 ∈ (V × V)) → 𝑤 ∈ (V × V))
12 elvv 5345 . . . . . . . . 9 (𝑤 ∈ (V × V) ↔ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
13 eleq1 2832 . . . . . . . . . . . . . 14 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹))
14 vex 3353 . . . . . . . . . . . . . . 15 𝑥 ∈ V
15 vex 3353 . . . . . . . . . . . . . . 15 𝑦 ∈ V
1614, 15opelcnv 5472 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹)
1713, 16syl6bb 278 . . . . . . . . . . . . 13 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤dom tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹))
18 sneq 4344 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
1918cnveqd 5466 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
2019unieqd 4604 . . . . . . . . . . . . . . 15 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
21 opswap 5808 . . . . . . . . . . . . . . 15 {⟨𝑥, 𝑦⟩} = ⟨𝑦, 𝑥
2220, 21syl6eq 2815 . . . . . . . . . . . . . 14 (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = ⟨𝑦, 𝑥⟩)
2322breq1d 4819 . . . . . . . . . . . . 13 (𝑤 = ⟨𝑥, 𝑦⟩ → ( {𝑤}tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
2417, 23anbi12d 624 . . . . . . . . . . . 12 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧)))
25 opex 5088 . . . . . . . . . . . . . . 15 𝑦, 𝑥⟩ ∈ V
26 vex 3353 . . . . . . . . . . . . . . 15 𝑧 ∈ V
2725, 26breldm 5497 . . . . . . . . . . . . . 14 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 → ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹)
2827pm4.71ri 556 . . . . . . . . . . . . 13 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
29 brtpos 7564 . . . . . . . . . . . . . 14 (𝑧 ∈ V → (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧))
3026, 29ax-mp 5 . . . . . . . . . . . . 13 (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧)
3128, 30bitr3i 268 . . . . . . . . . . . 12 ((⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧) ↔ ⟨𝑥, 𝑦𝐹𝑧)
3224, 31syl6bb 278 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ ⟨𝑥, 𝑦𝐹𝑧))
33 breq1 4812 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦𝐹𝑧))
3432, 33bitr4d 273 . . . . . . . . . 10 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
3534exlimivv 2027 . . . . . . . . 9 (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
3612, 35sylbi 208 . . . . . . . 8 (𝑤 ∈ (V × V) → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
37 iba 523 . . . . . . . 8 (𝑤 ∈ (V × V) → (𝑤𝐹𝑧 ↔ (𝑤𝐹𝑧𝑤 ∈ (V × V))))
3836, 37bitrd 270 . . . . . . 7 (𝑤 ∈ (V × V) → ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ (V × V))))
3910, 11, 38pm5.21nii 369 . . . . . 6 ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ (V × V)))
40 elsni 4351 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {∅} → 𝑤 = ∅)
4140sneqd 4346 . . . . . . . . . . . . . . 15 (𝑤 ∈ {∅} → {𝑤} = {∅})
4241cnveqd 5466 . . . . . . . . . . . . . 14 (𝑤 ∈ {∅} → {𝑤} = {∅})
43 cnvsn0 5786 . . . . . . . . . . . . . 14 {∅} = ∅
4442, 43syl6eq 2815 . . . . . . . . . . . . 13 (𝑤 ∈ {∅} → {𝑤} = ∅)
4544unieqd 4604 . . . . . . . . . . . 12 (𝑤 ∈ {∅} → {𝑤} = ∅)
46 uni0 4623 . . . . . . . . . . . 12 ∅ = ∅
4745, 46syl6eq 2815 . . . . . . . . . . 11 (𝑤 ∈ {∅} → {𝑤} = ∅)
4847breq1d 4819 . . . . . . . . . 10 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧 ↔ ∅tpos 𝐹𝑧))
49 brtpos0 7562 . . . . . . . . . . 11 (𝑧 ∈ V → (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
5026, 49ax-mp 5 . . . . . . . . . 10 (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧)
5148, 50syl6bb 278 . . . . . . . . 9 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
5240breq1d 4819 . . . . . . . . 9 (𝑤 ∈ {∅} → (𝑤𝐹𝑧 ↔ ∅𝐹𝑧))
5351, 52bitr4d 273 . . . . . . . 8 (𝑤 ∈ {∅} → ( {𝑤}tpos 𝐹𝑧𝑤𝐹𝑧))
5453pm5.32i 570 . . . . . . 7 ((𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧))
55 ancom 452 . . . . . . 7 ((𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ {∅}))
5654, 55bitri 266 . . . . . 6 ((𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤 ∈ {∅}))
5739, 56orbi12i 938 . . . . 5 (((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧)) ↔ ((𝑤𝐹𝑧𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧𝑤 ∈ {∅})))
58 andir 1031 . . . . 5 (((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ ((𝑤dom tpos 𝐹 {𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ {𝑤}tpos 𝐹𝑧)))
59 andi 1030 . . . . 5 ((𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})) ↔ ((𝑤𝐹𝑧𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧𝑤 ∈ {∅})))
6057, 58, 593bitr4i 294 . . . 4 (((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})))
61 elun 3915 . . . . 5 (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ↔ (𝑤dom tpos 𝐹𝑤 ∈ {∅}))
6261anbi1i 617 . . . 4 ((𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ ((𝑤dom tpos 𝐹𝑤 ∈ {∅}) ∧ {𝑤}tpos 𝐹𝑧))
63 brxp 5323 . . . . . . 7 (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 ∈ V))
6426, 63mpbiran2 701 . . . . . 6 (𝑤(((V × V) ∪ {∅}) × V)𝑧𝑤 ∈ ((V × V) ∪ {∅}))
65 elun 3915 . . . . . 6 (𝑤 ∈ ((V × V) ∪ {∅}) ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))
6664, 65bitri 266 . . . . 5 (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))
6766anbi2i 616 . . . 4 ((𝑤𝐹𝑧𝑤(((V × V) ∪ {∅}) × V)𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})))
6860, 62, 673bitr4i 294 . . 3 ((𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧𝑤(((V × V) ∪ {∅}) × V)𝑧))
69 brtpos2 7561 . . . 4 (𝑧 ∈ V → (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧)))
7026, 69ax-mp 5 . . 3 (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (dom tpos 𝐹 ∪ {∅}) ∧ {𝑤}tpos 𝐹𝑧))
71 brin 4861 . . 3 (𝑤(𝐹 ∩ (((V × V) ∪ {∅}) × V))𝑧 ↔ (𝑤𝐹𝑧𝑤(((V × V) ∪ {∅}) × V)𝑧))
7268, 70, 713bitr4i 294 . 2 (𝑤tpos tpos 𝐹𝑧𝑤(𝐹 ∩ (((V × V) ∪ {∅}) × V))𝑧)
731, 5, 72eqbrriv 5384 1 tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wo 873   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334  cop 4340   cuni 4594   class class class wbr 4809   × cxp 5275  ccnv 5276  dom cdm 5277  Rel wrel 5282  tpos ctpos 7554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076  df-tpos 7555
This theorem is referenced by:  tpostpos2  7576
  Copyright terms: Public domain W3C validator