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Theorem dftpos4 5800
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 5782 . . 3 tpos 𝐹 = (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x}))
2 relcnv 4630 . . . . . . 7 Rel dom 𝐹
3 df-rel 4279 . . . . . . 7 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
42, 3mpbi 133 . . . . . 6 dom 𝐹 ⊆ (V × V)
5 unss1 3089 . . . . . 6 (dom 𝐹 ⊆ (V × V) → (dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
6 resmpt 4583 . . . . . 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 4562 . . . . 5 ((x ((V × V) ∪ {∅}) ↦ {x}) ↾ (dom 𝐹 ∪ {∅})) ⊆ (x ((V × V) ∪ {∅}) ↦ {x})
97, 8eqsstr3i 2953 . . . 4 (x (dom 𝐹 ∪ {∅}) ↦ {x}) ⊆ (x ((V × V) ∪ {∅}) ↦ {x})
10 coss2 4419 . . . 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 2952 . 2 tpos 𝐹 ⊆ (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x}))
13 relco 4746 . . 3 Rel (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x}))
14 vex 2538 . . . . 5 y V
15 vex 2538 . . . . 5 z V
1614, 15opelco 4434 . . . 4 (⟨y, z (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x})) ↔ w(y(x ((V × V) ∪ {∅}) ↦ {x})w w𝐹z))
17 vex 2538 . . . . . . . . 9 w V
18 eleq1 2082 . . . . . . . . . 10 (x = y → (x ((V × V) ∪ {∅}) ↔ y ((V × V) ∪ {∅})))
19 sneq 3361 . . . . . . . . . . . . 13 (x = y → {x} = {y})
2019cnveqd 4438 . . . . . . . . . . . 12 (x = y{x} = {y})
2120unieqd 3565 . . . . . . . . . . 11 (x = y {x} = {y})
2221eqeq2d 2033 . . . . . . . . . 10 (x = y → (z = {x} ↔ z = {y}))
2318, 22anbi12d 445 . . . . . . . . 9 (x = y → ((x ((V × V) ∪ {∅}) z = {x}) ↔ (y ((V × V) ∪ {∅}) z = {y})))
24 eqeq1 2028 . . . . . . . . . 10 (z = w → (z = {y} ↔ w = {y}))
2524anbi2d 440 . . . . . . . . 9 (z = w → ((y ((V × V) ∪ {∅}) z = {y}) ↔ (y ((V × V) ∪ {∅}) w = {y})))
26 df-mpt 3794 . . . . . . . . 9 (x ((V × V) ∪ {∅}) ↦ {x}) = {⟨x, z⟩ ∣ (x ((V × V) ∪ {∅}) z = {x})}
2714, 17, 23, 25, 26brab 3983 . . . . . . . 8 (y(x ((V × V) ∪ {∅}) ↦ {x})w ↔ (y ((V × V) ∪ {∅}) w = {y}))
28 simplr 470 . . . . . . . . . . . 12 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → w = {y})
2917, 15breldm 4466 . . . . . . . . . . . . 13 (w𝐹zw dom 𝐹)
3029adantl 262 . . . . . . . . . . . 12 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → w dom 𝐹)
3128, 30eqeltrrd 2097 . . . . . . . . . . 11 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → {y} dom 𝐹)
32 elvv 4329 . . . . . . . . . . . . . 14 (y (V × V) ↔ zw y = ⟨z, w⟩)
33 opswapg 4734 . . . . . . . . . . . . . . . . . . 19 ((z V w V) → {⟨z, w⟩} = ⟨w, z⟩)
3415, 17, 33mp2an 404 . . . . . . . . . . . . . . . . . 18 {⟨z, w⟩} = ⟨w, z
3534eleq1i 2085 . . . . . . . . . . . . . . . . 17 ( {⟨z, w⟩} dom 𝐹 ↔ ⟨w, z dom 𝐹)
3615, 17opelcnv 4444 . . . . . . . . . . . . . . . . 17 (⟨z, w dom 𝐹 ↔ ⟨w, z dom 𝐹)
3735, 36bitr4i 176 . . . . . . . . . . . . . . . 16 ( {⟨z, w⟩} dom 𝐹 ↔ ⟨z, w dom 𝐹)
38 sneq 3361 . . . . . . . . . . . . . . . . . . . 20 (y = ⟨z, w⟩ → {y} = {⟨z, w⟩})
3938cnveqd 4438 . . . . . . . . . . . . . . . . . . 19 (y = ⟨z, w⟩ → {y} = {⟨z, w⟩})
4039unieqd 3565 . . . . . . . . . . . . . . . . . 18 (y = ⟨z, w⟩ → {y} = {⟨z, w⟩})
4140eleq1d 2088 . . . . . . . . . . . . . . . . 17 (y = ⟨z, w⟩ → ( {y} dom 𝐹 {⟨z, w⟩} dom 𝐹))
42 eleq1 2082 . . . . . . . . . . . . . . . . 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 1758 . . . . . . . . . . . . . 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 3087 . . . . . . . . . . . 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 3088 . . . . . . . . . . 11 (y {∅} → y (dom 𝐹 ∪ {∅}))
5251a1i 9 . . . . . . . . . 10 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → (y {∅} → y (dom 𝐹 ∪ {∅})))
53 simpll 469 . . . . . . . . . . 11 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → y ((V × V) ∪ {∅}))
54 elun 3061 . . . . . . . . . . 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 625 . . . . . . . . 9 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → y (dom 𝐹 ∪ {∅}))
57 simpr 103 . . . . . . . . . 10 (((y ((V × V) ∪ {∅}) w = {y}) w𝐹z) → w𝐹z)
5828, 57eqbrtrrd 3760 . . . . . . . . 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 5788 . . . . . . . 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 3739 . . . . . 6 (ytpos 𝐹z ↔ ⟨y, z tpos 𝐹)
6563, 64sylib 127 . . . . 5 ((y(x ((V × V) ∪ {∅}) ↦ {x})w w𝐹z) → ⟨y, z tpos 𝐹)
6665exlimiv 1471 . . . 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 4358 . 2 (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x})) ⊆ tpos 𝐹
6912, 68eqssi 2938 1 tpos 𝐹 = (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x}))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 616   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535  cun 2892  wss 2894  c0 3201  {csn 3350  cop 3353   cuni 3554   class class class wbr 3738  cmpt 3792   × cxp 4270  ccnv 4271  dom cdm 4272  cres 4274  ccom 4276  Rel wrel 4277  tpos ctpos 5781
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837  df-tpos 5782
This theorem is referenced by:  tposco  5812  nftpos  5816
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