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Theorem dftpos4 5819
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 5801 . . 3 tpos 𝐹 = (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x}))
2 relcnv 4646 . . . . . . 7 Rel dom 𝐹
3 df-rel 4295 . . . . . . 7 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
42, 3mpbi 133 . . . . . 6 dom 𝐹 ⊆ (V × V)
5 unss1 3106 . . . . . 6 (dom 𝐹 ⊆ (V × V) → (dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
6 resmpt 4599 . . . . . 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 4578 . . . . 5 ((x ((V × V) ∪ {∅}) ↦ {x}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (x ((V × V) ∪ {∅}) ↦ {x})
97, 8eqsstr3i 2970 . . . 4 (x (dom 𝐹 ∪ {∅}) ↦ {x}) ⊆ (x ((V × V) ∪ {∅}) ↦ {x})
10 coss2 4435 . . . 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 2969 . 2 tpos 𝐹 ⊆ (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x}))
13 relco 4762 . . 3 Rel (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x}))
14 vex 2554 . . . . 5 y V
15 vex 2554 . . . . 5 z V
1614, 15opelco 4450 . . . 4 (⟨y, z (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x})) ↔ w(y(x ((V × V) ∪ {∅}) ↦ {x})w w𝐹z))
17 vex 2554 . . . . . . . . 9 w V
18 eleq1 2097 . . . . . . . . . 10 (x = y → (x ((V × V) ∪ {∅}) ↔ y ((V × V) ∪ {∅})))
19 sneq 3378 . . . . . . . . . . . . 13 (x = y → {x} = {y})
2019cnveqd 4454 . . . . . . . . . . . 12 (x = y{x} = {y})
2120unieqd 3582 . . . . . . . . . . 11 (x = y {x} = {y})
2221eqeq2d 2048 . . . . . . . . . 10 (x = y → (z = {x} ↔ z = {y}))
2318, 22anbi12d 442 . . . . . . . . 9 (x = y → ((x ((V × V) ∪ {∅}) z = {x}) ↔ (y ((V × V) ∪ {∅}) z = {y})))
24 eqeq1 2043 . . . . . . . . . 10 (z = w → (z = {y} ↔ w = {y}))
2524anbi2d 437 . . . . . . . . 9 (z = w → ((y ((V × V) ∪ {∅}) z = {y}) ↔ (y ((V × V) ∪ {∅}) w = {y})))
26 df-mpt 3811 . . . . . . . . 9 (x ((V × V) ∪ {∅}) ↦ {x}) = {⟨x, z⟩ ∣ (x ((V × V) ∪ {∅}) z = {x})}
2714, 17, 23, 25, 26brab 4000 . . . . . . . 8 (y(x ((V × V) ∪ {∅}) ↦ {x})w ↔ (y ((V × V) ∪ {∅}) w = {y}))
28 simplr 482 . . . . . . . . . . . 12 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → w = {y})
2917, 15breldm 4482 . . . . . . . . . . . . 13 (w𝐹zw dom 𝐹)
3029adantl 262 . . . . . . . . . . . 12 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → w dom 𝐹)
3128, 30eqeltrrd 2112 . . . . . . . . . . 11 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → {y} dom 𝐹)
32 elvv 4345 . . . . . . . . . . . . . 14 (y (V × V) ↔ zw y = ⟨z, w⟩)
33 opswapg 4750 . . . . . . . . . . . . . . . . . . 19 ((z V w V) → {⟨z, w⟩} = ⟨w, z⟩)
3415, 17, 33mp2an 402 . . . . . . . . . . . . . . . . . 18 {⟨z, w⟩} = ⟨w, z
3534eleq1i 2100 . . . . . . . . . . . . . . . . 17 ( {⟨z, w⟩} dom 𝐹 ↔ ⟨w, z dom 𝐹)
3615, 17opelcnv 4460 . . . . . . . . . . . . . . . . 17 (⟨z, w dom 𝐹 ↔ ⟨w, z dom 𝐹)
3735, 36bitr4i 176 . . . . . . . . . . . . . . . 16 ( {⟨z, w⟩} dom 𝐹 ↔ ⟨z, w dom 𝐹)
38 sneq 3378 . . . . . . . . . . . . . . . . . . . 20 (y = ⟨z, w⟩ → {y} = {⟨z, w⟩})
3938cnveqd 4454 . . . . . . . . . . . . . . . . . . 19 (y = ⟨z, w⟩ → {y} = {⟨z, w⟩})
4039unieqd 3582 . . . . . . . . . . . . . . . . . 18 (y = ⟨z, w⟩ → {y} = {⟨z, w⟩})
4140eleq1d 2103 . . . . . . . . . . . . . . . . 17 (y = ⟨z, w⟩ → ( {y} dom 𝐹 {⟨z, w⟩} dom 𝐹))
42 eleq1 2097 . . . . . . . . . . . . . . . . 17 (y = ⟨z, w⟩ → (y dom 𝐹 ↔ ⟨z, w dom 𝐹))
4341, 42bibi12d 224 . . . . . . . . . . . . . . . 16 (y = ⟨z, w⟩ → (( {y} dom 𝐹y dom 𝐹) ↔ ( {⟨z, w⟩} dom 𝐹 ↔ ⟨z, w dom 𝐹)))
4437, 43mpbiri 157 . . . . . . . . . . . . . . 15 (y = ⟨z, w⟩ → ( {y} dom 𝐹y dom 𝐹))
4544exlimivv 1773 . . . . . . . . . . . . . 14 (zw y = ⟨z, w⟩ → ( {y} dom 𝐹y dom 𝐹))
4632, 45sylbi 114 . . . . . . . . . . . . 13 (y (V × V) → ( {y} dom 𝐹y dom 𝐹))
4746biimpcd 148 . . . . . . . . . . . 12 ( {y} dom 𝐹 → (y (V × V) → y dom 𝐹))
48 elun1 3104 . . . . . . . . . . . 12 (y dom 𝐹y (dom 𝐹 ∪ {∅}))
4947, 48syl6 29 . . . . . . . . . . 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 3105 . . . . . . . . . . 11 (y {∅} → y (dom 𝐹 ∪ {∅}))
5251a1i 9 . . . . . . . . . 10 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → (y {∅} → y (dom 𝐹 ∪ {∅})))
53 simpll 481 . . . . . . . . . . 11 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → y ((V × V) ∪ {∅}))
54 elun 3078 . . . . . . . . . . 11 (y ((V × V) ∪ {∅}) ↔ (y (V × V) y {∅}))
5553, 54sylib 127 . . . . . . . . . 10 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → (y (V × V) y {∅}))
5650, 52, 55mpjaod 637 . . . . . . . . 9 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → y (dom 𝐹 ∪ {∅}))
57 simpr 103 . . . . . . . . . 10 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → w𝐹z)
5828, 57eqbrtrrd 3777 . . . . . . . . 9 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → {y}𝐹z)
5956, 58jca 290 . . . . . . . 8 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → (y (dom 𝐹 ∪ {∅}) {y}𝐹z))
6027, 59sylanb 268 . . . . . . 7 ((y(x ((V × V) ∪ {∅}) ↦ {x})w w𝐹z) → (y (dom 𝐹 ∪ {∅}) {y}𝐹z))
61 brtpos2 5807 . . . . . . . 8 (z V → (ytpos 𝐹z ↔ (y (dom 𝐹 ∪ {∅}) {y}𝐹z)))
6215, 61ax-mp 7 . . . . . . 7 (ytpos 𝐹z ↔ (y (dom 𝐹 ∪ {∅}) {y}𝐹z))
6360, 62sylibr 137 . . . . . 6 ((y(x ((V × V) ∪ {∅}) ↦ {x})w w𝐹z) → ytpos 𝐹z)
64 df-br 3756 . . . . . 6 (ytpos 𝐹z ↔ ⟨y, z tpos 𝐹)
6563, 64sylib 127 . . . . 5 ((y(x ((V × V) ∪ {∅}) ↦ {x})w w𝐹z) → ⟨y, z tpos 𝐹)
6665exlimiv 1486 . . . 4 (w(y(x ((V × V) ∪ {∅}) ↦ {x})w w𝐹z) → ⟨y, z tpos 𝐹)
6716, 66sylbi 114 . . 3 (⟨y, z (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x})) → ⟨y, z tpos 𝐹)
6813, 67relssi 4374 . 2 (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x})) ⊆ tpos 𝐹
6912, 68eqssi 2955 1 tpos 𝐹 = (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x}))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cun 2909  wss 2911  c0 3218  {csn 3367  cop 3370   cuni 3571   class class class wbr 3755  cmpt 3809   × cxp 4286  ccnv 4287  dom cdm 4288  cres 4290  ccom 4292  Rel wrel 4293  tpos ctpos 5800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-tpos 5801
This theorem is referenced by:  tposco  5831  nftpos  5835
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