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Theorem dftpos3 5907
Description: Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4380. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
dftpos3 (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧})
Distinct variable group:   𝑥,𝑦,𝑧,𝐹

Proof of Theorem dftpos3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 relcnv 4730 . . . . . . . . . 10 Rel dom 𝐹
2 dmtpos 5901 . . . . . . . . . . 11 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
32releqd 4451 . . . . . . . . . 10 (Rel dom 𝐹 → (Rel dom tpos 𝐹 ↔ Rel dom 𝐹))
41, 3mpbiri 161 . . . . . . . . 9 (Rel dom 𝐹 → Rel dom tpos 𝐹)
5 reltpos 5895 . . . . . . . . 9 Rel tpos 𝐹
64, 5jctil 299 . . . . . . . 8 (Rel dom 𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹))
7 relrelss 4871 . . . . . . . 8 ((Rel tpos 𝐹 ∧ Rel dom tpos 𝐹) ↔ tpos 𝐹 ⊆ ((V × V) × V))
86, 7sylib 131 . . . . . . 7 (Rel dom 𝐹 → tpos 𝐹 ⊆ ((V × V) × V))
98sseld 2971 . . . . . 6 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹𝑤 ∈ ((V × V) × V)))
10 elvvv 4430 . . . . . 6 (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
119, 10syl6ib 154 . . . . 5 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → ∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
1211pm4.71rd 380 . . . 4 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹)))
13 19.41vvv 1800 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹))
14 eleq1 2116 . . . . . . . 8 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹))
15 df-br 3792 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹)
16 vex 2577 . . . . . . . . . 10 𝑥 ∈ V
17 vex 2577 . . . . . . . . . 10 𝑦 ∈ V
18 vex 2577 . . . . . . . . . 10 𝑧 ∈ V
19 brtposg 5899 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2016, 17, 18, 19mp3an 1243 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧)
2115, 20bitr3i 179 . . . . . . . 8 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥𝐹𝑧)
2214, 21syl6bb 189 . . . . . . 7 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2322pm5.32i 435 . . . . . 6 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
24233exbii 1514 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
2513, 24bitr3i 179 . . . 4 ((∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
2612, 25syl6bb 189 . . 3 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)))
2726abbi2dv 2172 . 2 (Rel dom 𝐹 → tpos 𝐹 = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)})
28 df-oprab 5543 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)}
2927, 28syl6eqr 2106 1 (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  {cab 2042  Vcvv 2574  wss 2944  cop 3405   class class class wbr 3791   × cxp 4370  ccnv 4371  dom cdm 4372  Rel wrel 4377  {coprab 5540  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-in1 554  ax-in2 555  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-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  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-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  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-oprab 5543  df-tpos 5890
This theorem is referenced by:  tposoprab  5925
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