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Theorem dftpos3 7316
Description: Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 5087. (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 5466 . . . . . . . . . 10 Rel dom 𝐹
2 dmtpos 7310 . . . . . . . . . . 11 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
32releqd 5169 . . . . . . . . . 10 (Rel dom 𝐹 → (Rel dom tpos 𝐹 ↔ Rel dom 𝐹))
41, 3mpbiri 248 . . . . . . . . 9 (Rel dom 𝐹 → Rel dom tpos 𝐹)
5 reltpos 7303 . . . . . . . . 9 Rel tpos 𝐹
64, 5jctil 559 . . . . . . . 8 (Rel dom 𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹))
7 relrelss 5621 . . . . . . . 8 ((Rel tpos 𝐹 ∧ Rel dom tpos 𝐹) ↔ tpos 𝐹 ⊆ ((V × V) × V))
86, 7sylib 208 . . . . . . 7 (Rel dom 𝐹 → tpos 𝐹 ⊆ ((V × V) × V))
98sseld 3587 . . . . . 6 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹𝑤 ∈ ((V × V) × V)))
10 elvvv 5144 . . . . . 6 (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
119, 10syl6ib 241 . . . . 5 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → ∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
1211pm4.71rd 666 . . . 4 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹)))
13 19.41vvv 1918 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹))
14 eleq1 2692 . . . . . . . 8 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹))
15 df-br 4619 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹)
16 vex 3194 . . . . . . . . . 10 𝑧 ∈ V
17 brtpos 7307 . . . . . . . . . 10 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
1816, 17ax-mp 5 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧)
1915, 18bitr3i 266 . . . . . . . 8 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥𝐹𝑧)
2014, 19syl6bb 276 . . . . . . 7 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2120pm5.32i 668 . . . . . 6 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
22213exbii 1774 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
2313, 22bitr3i 266 . . . 4 ((∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
2412, 23syl6bb 276 . . 3 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)))
2524abbi2dv 2745 . 2 (Rel dom 𝐹 → tpos 𝐹 = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)})
26 df-oprab 6609 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)}
2725, 26syl6eqr 2678 1 (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992  {cab 2612  Vcvv 3191  wss 3560  cop 4159   class class class wbr 4618   × cxp 5077  ccnv 5078  dom cdm 5079  Rel wrel 5084  {coprab 6606  tpos ctpos 7297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-fv 5858  df-oprab 6609  df-tpos 7298
This theorem is referenced by:  tposoprab  7334
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