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Theorem dftpos3 8231
Description: Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 5683. (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 6102 . . . . . . . . . 10 Rel dom 𝐹
2 dmtpos 8225 . . . . . . . . . . 11 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
32releqd 5777 . . . . . . . . . 10 (Rel dom 𝐹 → (Rel dom tpos 𝐹 ↔ Rel dom 𝐹))
41, 3mpbiri 257 . . . . . . . . 9 (Rel dom 𝐹 → Rel dom tpos 𝐹)
5 reltpos 8218 . . . . . . . . 9 Rel tpos 𝐹
64, 5jctil 518 . . . . . . . 8 (Rel dom 𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹))
7 relrelss 6271 . . . . . . . 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 3980 . . . . . 6 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹𝑤 ∈ ((V × V) × V)))
10 elvvv 5750 . . . . . 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 561 . . . 4 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹)))
13 19.41vvv 1953 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹))
14 eleq1 2819 . . . . . . . 8 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹))
15 df-br 5148 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹)
16 brtpos 8222 . . . . . . . . . 10 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
1716elv 3478 . . . . . . . . 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 573 . . . . . 6 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
21203exbii 1850 . . . . 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 2865 . 2 (Rel dom 𝐹 → tpos 𝐹 = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)})
25 df-oprab 7415 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)}
2624, 25eqtr4di 2788 1 (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  {cab 2707  Vcvv 3472  wss 3947  cop 4633   class class class wbr 5147   × cxp 5673  ccnv 5674  dom cdm 5675  Rel wrel 5680  {coprab 7412  tpos ctpos 8212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-oprab 7415  df-tpos 8213
This theorem is referenced by:  tposoprab  8249
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