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Theorem dftpos4 5908
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 5890 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
2 relcnv 4730 . . . . . . 7 Rel dom 𝐹
3 df-rel 4379 . . . . . . 7 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
42, 3mpbi 137 . . . . . 6 dom 𝐹 ⊆ (V × V)
5 unss1 3139 . . . . . 6 (dom 𝐹 ⊆ (V × V) → (dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
6 resmpt 4683 . . . . . 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 4662 . . . . 5 ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
97, 8eqsstr3i 3003 . . . 4 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
10 coss2 4519 . . . 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 3002 . 2 tpos 𝐹 ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
13 relco 4846 . . 3 Rel (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
14 vex 2577 . . . . 5 𝑦 ∈ V
15 vex 2577 . . . . 5 𝑧 ∈ V
1614, 15opelco 4534 . . . 4 (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ↔ ∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧))
17 vex 2577 . . . . . . . . 9 𝑤 ∈ V
18 eleq1 2116 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ((V × V) ∪ {∅}) ↔ 𝑦 ∈ ((V × V) ∪ {∅})))
19 sneq 3413 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2019cnveqd 4538 . . . . . . . . . . . 12 (𝑥 = 𝑦{𝑥} = {𝑦})
2120unieqd 3618 . . . . . . . . . . 11 (𝑥 = 𝑦 {𝑥} = {𝑦})
2221eqeq2d 2067 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑧 = {𝑥} ↔ 𝑧 = {𝑦}))
2318, 22anbi12d 450 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑥}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑦})))
24 eqeq1 2062 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑧 = {𝑦} ↔ 𝑤 = {𝑦}))
2524anbi2d 445 . . . . . . . . 9 (𝑧 = 𝑤 → ((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑦}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦})))
26 df-mpt 3847 . . . . . . . . 9 (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = {𝑥})}
2714, 17, 23, 25, 26brab 4036 . . . . . . . 8 (𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤 ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}))
28 simplr 490 . . . . . . . . . . . 12 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 = {𝑦})
2917, 15breldm 4566 . . . . . . . . . . . . 13 (𝑤𝐹𝑧𝑤 ∈ dom 𝐹)
3029adantl 266 . . . . . . . . . . . 12 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 ∈ dom 𝐹)
3128, 30eqeltrrd 2131 . . . . . . . . . . 11 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → {𝑦} ∈ dom 𝐹)
32 elvv 4429 . . . . . . . . . . . . . 14 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
33 opswapg 4834 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → {⟨𝑧, 𝑤⟩} = ⟨𝑤, 𝑧⟩)
3415, 17, 33mp2an 410 . . . . . . . . . . . . . . . . . 18 {⟨𝑧, 𝑤⟩} = ⟨𝑤, 𝑧
3534eleq1i 2119 . . . . . . . . . . . . . . . . 17 ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
3615, 17opelcnv 4544 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
3735, 36bitr4i 180 . . . . . . . . . . . . . . . 16 ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹)
38 sneq 3413 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
3938cnveqd 4538 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
4039unieqd 3618 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
4140eleq1d 2122 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹 {⟨𝑧, 𝑤⟩} ∈ dom 𝐹))
42 eleq1 2116 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹))
4341, 42bibi12d 228 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, 𝑤⟩ → (( {𝑦} ∈ dom 𝐹𝑦dom 𝐹) ↔ ( {⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ dom 𝐹)))
4437, 43mpbiri 161 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4544exlimivv 1792 . . . . . . . . . . . . . 14 (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4632, 45sylbi 118 . . . . . . . . . . . . 13 (𝑦 ∈ (V × V) → ( {𝑦} ∈ dom 𝐹𝑦dom 𝐹))
4746biimpcd 152 . . . . . . . . . . . 12 ( {𝑦} ∈ dom 𝐹 → (𝑦 ∈ (V × V) → 𝑦dom 𝐹))
48 elun1 3137 . . . . . . . . . . . 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 3138 . . . . . . . . . . 11 (𝑦 ∈ {∅} → 𝑦 ∈ (dom 𝐹 ∪ {∅}))
5251a1i 9 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ {∅} → 𝑦 ∈ (dom 𝐹 ∪ {∅})))
53 simpll 489 . . . . . . . . . . 11 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ ((V × V) ∪ {∅}))
54 elun 3111 . . . . . . . . . . 11 (𝑦 ∈ ((V × V) ∪ {∅}) ↔ (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
5553, 54sylib 131 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
5650, 52, 55mpjaod 648 . . . . . . . . 9 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ (dom 𝐹 ∪ {∅}))
57 simpr 107 . . . . . . . . . 10 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤𝐹𝑧)
5828, 57eqbrtrrd 3813 . . . . . . . . 9 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → {𝑦}𝐹𝑧)
5956, 58jca 294 . . . . . . . 8 (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = {𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
6027, 59sylanb 272 . . . . . . 7 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
61 brtpos2 5896 . . . . . . . 8 (𝑧 ∈ V → (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧)))
6215, 61ax-mp 7 . . . . . . 7 (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑦}𝐹𝑧))
6360, 62sylibr 141 . . . . . 6 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → 𝑦tpos 𝐹𝑧)
64 df-br 3792 . . . . . 6 (𝑦tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6563, 64sylib 131 . . . . 5 ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6665exlimiv 1505 . . . 4 (∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})𝑤𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6716, 66sylbi 118 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
6813, 67relssi 4458 . 2 (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ⊆ tpos 𝐹
6912, 68eqssi 2988 1 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 639   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  cun 2942  wss 2944  c0 3251  {csn 3402  cop 3405   cuni 3607   class class class wbr 3791  cmpt 3845   × cxp 4370  ccnv 4371  dom cdm 4372  cres 4374  ccom 4376  Rel wrel 4377  tpos ctpos 5889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937  df-tpos 5890
This theorem is referenced by:  tposco  5920  nftpos  5924
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