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Theorem dftpos4 5984
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 5966 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
2 relcnv 4779 . . . . . . 7 Rel dom 𝐹
3 df-rel 4420 . . . . . . 7 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
42, 3mpbi 143 . . . . . 6 dom 𝐹 ⊆ (V × V)
5 unss1 3158 . . . . . 6 (dom 𝐹 ⊆ (V × V) → (dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
6 resmpt 4729 . . . . . 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 4706 . . . . 5 ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
97, 8eqsstr3i 3046 . . . 4 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
10 coss2 4562 . . . 4 ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) → (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
119, 10ax-mp 7 . . 3 (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
121, 11eqsstri 3045 . 2 tpos 𝐹 ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
13 relco 4897 . . 3 Rel (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
14 vex 2618 . . . . 5 𝑦 ∈ V
15 vex 2618 . . . . 5 𝑧 ∈ V
1614, 15opelco 4578 . . . 4 (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ↔ ∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧))
17 vex 2618 . . . . . . . . 9 𝑤 ∈ V
18 eleq1 2147 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ((V × V) ∪ {∅}) ↔ 𝑦 ∈ ((V × V) ∪ {∅})))
19 sneq 3442 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2019cnveqd 4582 . . . . . . . . . . . 12 (𝑥 = 𝑦{𝑥} = {𝑦})
2120unieqd 3649 . . . . . . . . . . 11 (𝑥 = 𝑦 {𝑥} = {𝑦})
2221eqeq2d 2096 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑧 = {𝑥} ↔ 𝑧 = {𝑦}))
2318, 22anbi12d 457 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑥}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑦})))
24 eqeq1 2091 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑧 = {𝑦} ↔ 𝑤 = {𝑦}))
2524anbi2d 452 . . . . . . . . 9 (𝑧 = 𝑤 → ((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑦}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦})))
26 df-mpt 3878 . . . . . . . . 9 (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑥})}
2714, 17, 23, 25, 26brab 4075 . . . . . . . 8 (𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤 ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}))
28 simplr 497 . . . . . . . . . . . 12 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 = {𝑦})
2917, 15breldm 4610 . . . . . . . . . . . . 13 (𝑤𝐹𝑧𝑤 ∈ dom 𝐹)
3029adantl 271 . . . . . . . . . . . 12 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 ∈ dom 𝐹)
3128, 30eqeltrrd 2162 . . . . . . . . . . 11 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → {𝑦} ∈ dom 𝐹)
32 elvv 4470 . . . . . . . . . . . . . 14 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
33 opswapg 4885 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → {⟨𝑧, 𝑤⟩} = ⟨𝑤, 𝑧⟩)
3415, 17, 33mp2an 417 . . . . . . . . . . . . . . . . . 18 {⟨𝑧, 𝑤⟩} = ⟨𝑤, 𝑧
3534eleq1i 2150 . . . . . . . . . . . . . . . . 17 ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
3615, 17opelcnv 4588 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
3735, 36bitr4i 185 . . . . . . . . . . . . . . . 16 ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹)
38 sneq 3442 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
3938cnveqd 4582 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
4039unieqd 3649 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
4140eleq1d 2153 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹 {⟨𝑧, 𝑤⟩} ∈ dom 𝐹))
42 eleq1 2147 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹))
4341, 42bibi12d 233 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, 𝑤⟩ → (( {𝑦} ∈ dom 𝐹𝑦dom 𝐹) ↔ ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹)))
4437, 43mpbiri 166 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4544exlimivv 1821 . . . . . . . . . . . . . 14 (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4632, 45sylbi 119 . . . . . . . . . . . . 13 (𝑦 ∈ (V × V) → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4746biimpcd 157 . . . . . . . . . . . 12 ( {𝑦} ∈ dom 𝐹 → (𝑦 ∈ (V × V) → 𝑦dom 𝐹))
48 elun1 3156 . . . . . . . . . . . 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 3157 . . . . . . . . . . 11 (𝑦 ∈ {∅} → 𝑦 ∈ (dom 𝐹 ∪ {∅}))
5251a1i 9 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ {∅} → 𝑦 ∈ (dom 𝐹 ∪ {∅})))
53 simpll 496 . . . . . . . . . . 11 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ ((V × V) ∪ {∅}))
54 elun 3130 . . . . . . . . . . 11 (𝑦 ∈ ((V × V) ∪ {∅}) ↔ (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
5553, 54sylib 120 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
5650, 52, 55mpjaod 671 . . . . . . . . 9 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ (dom 𝐹 ∪ {∅}))
57 simpr 108 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤𝐹𝑧)
5828, 57eqbrtrrd 3844 . . . . . . . . 9 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → {𝑦}𝐹𝑧)
5956, 58jca 300 . . . . . . . 8 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
6027, 59sylanb 278 . . . . . . 7 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
61 brtpos2 5972 . . . . . . . 8 (𝑧 ∈ V → (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧)))
6215, 61ax-mp 7 . . . . . . 7 (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
6360, 62sylibr 132 . . . . . 6 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → 𝑦tpos 𝐹𝑧)
64 df-br 3823 . . . . . 6 (𝑦tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6563, 64sylib 120 . . . . 5 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6665exlimiv 1532 . . . 4 (∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6716, 66sylbi 119 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6813, 67relssi 4499 . 2 (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ⊆ tpos 𝐹
6912, 68eqssi 3030 1 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662   = wceq 1287  wex 1424  wcel 1436  Vcvv 2615  cun 2986  wss 2988  c0 3275  {csn 3431  cop 3434   cuni 3638   class class class wbr 3822  cmpt 3876   × cxp 4411  ccnv 4412  dom cdm 4413  cres 4415  ccom 4417  Rel wrel 4418  tpos ctpos 5965
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-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-fv 4991  df-tpos 5966
This theorem is referenced by:  tposco  5996  nftpos  6000
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