Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dftpos4 GIF version

Theorem dftpos4 6160
 Description: Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
dftpos4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
Distinct variable group:   𝑥,𝐹

Proof of Theorem dftpos4
Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-tpos 6142 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
2 relcnv 4917 . . . . . . 7 Rel dom 𝐹
3 df-rel 4546 . . . . . . 7 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
42, 3mpbi 144 . . . . . 6 dom 𝐹 ⊆ (V × V)
5 unss1 3245 . . . . . 6 (dom 𝐹 ⊆ (V × V) → (dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
6 resmpt 4867 . . . . . 6 ((dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}) → ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
74, 5, 6mp2b 8 . . . . 5 ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
8 resss 4843 . . . . 5 ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
97, 8eqsstrri 3130 . . . 4 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
10 coss2 4695 . . . 4 ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
119, 10ax-mp 5 . . 3 (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
121, 11eqsstri 3129 . 2 tpos 𝐹 ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
13 relco 5037 . . 3 Rel (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
14 vex 2689 . . . . 5 𝑦 ∈ V
15 vex 2689 . . . . 5 𝑧 ∈ V
1614, 15opelco 4711 . . . 4 (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ↔ ∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧))
17 vex 2689 . . . . . . . . 9 𝑤 ∈ V
18 eleq1 2202 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ((V × V) ∪ {∅}) ↔ 𝑦 ∈ ((V × V) ∪ {∅})))
19 sneq 3538 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2019cnveqd 4715 . . . . . . . . . . . 12 (𝑥 = 𝑦{𝑥} = {𝑦})
2120unieqd 3747 . . . . . . . . . . 11 (𝑥 = 𝑦 {𝑥} = {𝑦})
2221eqeq2d 2151 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑧 = {𝑥} ↔ 𝑧 = {𝑦}))
2318, 22anbi12d 464 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑥}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑦})))
24 eqeq1 2146 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑧 = {𝑦} ↔ 𝑤 = {𝑦}))
2524anbi2d 459 . . . . . . . . 9 (𝑧 = 𝑤 → ((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑦}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦})))
26 df-mpt 3991 . . . . . . . . 9 (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑥})}
2714, 17, 23, 25, 26brab 4194 . . . . . . . 8 (𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤 ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}))
28 simplr 519 . . . . . . . . . . . 12 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 = {𝑦})
2917, 15breldm 4743 . . . . . . . . . . . . 13 (𝑤𝐹𝑧𝑤 ∈ dom 𝐹)
3029adantl 275 . . . . . . . . . . . 12 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 ∈ dom 𝐹)
3128, 30eqeltrrd 2217 . . . . . . . . . . 11 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → {𝑦} ∈ dom 𝐹)
32 elvv 4601 . . . . . . . . . . . . . 14 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
33 opswapg 5025 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → {⟨𝑧, 𝑤⟩} = ⟨𝑤, 𝑧⟩)
3415, 17, 33mp2an 422 . . . . . . . . . . . . . . . . . 18 {⟨𝑧, 𝑤⟩} = ⟨𝑤, 𝑧
3534eleq1i 2205 . . . . . . . . . . . . . . . . 17 ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
3615, 17opelcnv 4721 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
3735, 36bitr4i 186 . . . . . . . . . . . . . . . 16 ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹)
38 sneq 3538 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
3938cnveqd 4715 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
4039unieqd 3747 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
4140eleq1d 2208 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹 {⟨𝑧, 𝑤⟩} ∈ dom 𝐹))
42 eleq1 2202 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹))
4341, 42bibi12d 234 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, 𝑤⟩ → (( {𝑦} ∈ dom 𝐹𝑦dom 𝐹) ↔ ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹)))
4437, 43mpbiri 167 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4544exlimivv 1868 . . . . . . . . . . . . . 14 (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4632, 45sylbi 120 . . . . . . . . . . . . 13 (𝑦 ∈ (V × V) → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4746biimpcd 158 . . . . . . . . . . . 12 ( {𝑦} ∈ dom 𝐹 → (𝑦 ∈ (V × V) → 𝑦dom 𝐹))
48 elun1 3243 . . . . . . . . . . . 12 (𝑦dom 𝐹𝑦 ∈ (dom 𝐹 ∪ {∅}))
4947, 48syl6 33 . . . . . . . . . . 11 ( {𝑦} ∈ dom 𝐹 → (𝑦 ∈ (V × V) → 𝑦 ∈ (dom 𝐹 ∪ {∅})))
5031, 49syl 14 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (V × V) → 𝑦 ∈ (dom 𝐹 ∪ {∅})))
51 elun2 3244 . . . . . . . . . . 11 (𝑦 ∈ {∅} → 𝑦 ∈ (dom 𝐹 ∪ {∅}))
5251a1i 9 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ {∅} → 𝑦 ∈ (dom 𝐹 ∪ {∅})))
53 simpll 518 . . . . . . . . . . 11 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ ((V × V) ∪ {∅}))
54 elun 3217 . . . . . . . . . . 11 (𝑦 ∈ ((V × V) ∪ {∅}) ↔ (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
5553, 54sylib 121 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
5650, 52, 55mpjaod 707 . . . . . . . . 9 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ (dom 𝐹 ∪ {∅}))
57 simpr 109 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤𝐹𝑧)
5828, 57eqbrtrrd 3952 . . . . . . . . 9 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → {𝑦}𝐹𝑧)
5956, 58jca 304 . . . . . . . 8 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
6027, 59sylanb 282 . . . . . . 7 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
61 brtpos2 6148 . . . . . . . 8 (𝑧 ∈ V → (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧)))
6215, 61ax-mp 5 . . . . . . 7 (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
6360, 62sylibr 133 . . . . . 6 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → 𝑦tpos 𝐹𝑧)
64 df-br 3930 . . . . . 6 (𝑦tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6563, 64sylib 121 . . . . 5 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6665exlimiv 1577 . . . 4 (∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6716, 66sylbi 120 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6813, 67relssi 4630 . 2 (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ⊆ tpos 𝐹
6912, 68eqssi 3113 1 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686   ∪ cun 3069   ⊆ wss 3071  ∅c0 3363  {csn 3527  ⟨cop 3530  ∪ cuni 3736   class class class wbr 3929   ↦ cmpt 3989   × cxp 4537  ◡ccnv 4538  dom cdm 4539   ↾ cres 4541   ∘ ccom 4543  Rel wrel 4544  tpos ctpos 6141 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-tpos 6142 This theorem is referenced by:  tposco  6172  nftpos  6176
 Copyright terms: Public domain W3C validator