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Theorem dftpos3 8228
Description: Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 5684. (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 6103 . . . . . . . . . 10 Rel dom 𝐹
2 dmtpos 8222 . . . . . . . . . . 11 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
32releqd 5778 . . . . . . . . . 10 (Rel dom 𝐹 → (Rel dom tpos 𝐹 ↔ Rel dom 𝐹))
41, 3mpbiri 257 . . . . . . . . 9 (Rel dom 𝐹 → Rel dom tpos 𝐹)
5 reltpos 8215 . . . . . . . . 9 Rel tpos 𝐹
64, 5jctil 520 . . . . . . . 8 (Rel dom 𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹))
7 relrelss 6272 . . . . . . . 8 ((Rel tpos 𝐹 ∧ Rel dom tpos 𝐹) ↔ tpos 𝐹 ⊆ ((V × V) × V))
86, 7sylib 217 . . . . . . 7 (Rel dom 𝐹 → tpos 𝐹 ⊆ ((V × V) × V))
98sseld 3981 . . . . . 6 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹𝑤 ∈ ((V × V) × V)))
10 elvvv 5751 . . . . . 6 (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
119, 10imbitrdi 250 . . . . 5 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → ∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
1211pm4.71rd 563 . . . 4 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹)))
13 19.41vvv 1955 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹))
14 eleq1 2821 . . . . . . . 8 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹))
15 df-br 5149 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹)
16 brtpos 8219 . . . . . . . . . 10 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
1716elv 3480 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧)
1815, 17bitr3i 276 . . . . . . . 8 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥𝐹𝑧)
1914, 18bitrdi 286 . . . . . . 7 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2019pm5.32i 575 . . . . . 6 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
21203exbii 1852 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
2213, 21bitr3i 276 . . . 4 ((∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
2312, 22bitrdi 286 . . 3 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)))
2423eqabdv 2867 . 2 (Rel dom 𝐹 → tpos 𝐹 = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)})
25 df-oprab 7412 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)}
2624, 25eqtr4di 2790 1 (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2709  Vcvv 3474  wss 3948  cop 4634   class class class wbr 5148   × cxp 5674  ccnv 5675  dom cdm 5676  Rel wrel 5681  {coprab 7409  tpos ctpos 8209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-oprab 7412  df-tpos 8210
This theorem is referenced by:  tposoprab  8246
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