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Theorem dftpos3 6256
Description: Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4630. (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 5001 . . . . . . . . . 10 Rel dom 𝐹
2 dmtpos 6250 . . . . . . . . . . 11 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
32releqd 4706 . . . . . . . . . 10 (Rel dom 𝐹 → (Rel dom tpos 𝐹 ↔ Rel dom 𝐹))
41, 3mpbiri 168 . . . . . . . . 9 (Rel dom 𝐹 → Rel dom tpos 𝐹)
5 reltpos 6244 . . . . . . . . 9 Rel tpos 𝐹
64, 5jctil 312 . . . . . . . 8 (Rel dom 𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹))
7 relrelss 5150 . . . . . . . 8 ((Rel tpos 𝐹 ∧ Rel dom tpos 𝐹) ↔ tpos 𝐹 ⊆ ((V × V) × V))
86, 7sylib 122 . . . . . . 7 (Rel dom 𝐹 → tpos 𝐹 ⊆ ((V × V) × V))
98sseld 3154 . . . . . 6 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹𝑤 ∈ ((V × V) × V)))
10 elvvv 4685 . . . . . 6 (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
119, 10syl6ib 161 . . . . 5 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → ∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
1211pm4.71rd 394 . . . 4 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹)))
13 19.41vvv 1904 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹))
14 eleq1 2240 . . . . . . . 8 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹))
15 df-br 4001 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹)
16 vex 2740 . . . . . . . . . 10 𝑥 ∈ V
17 vex 2740 . . . . . . . . . 10 𝑦 ∈ V
18 vex 2740 . . . . . . . . . 10 𝑧 ∈ V
19 brtposg 6248 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2016, 17, 18, 19mp3an 1337 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧)
2115, 20bitr3i 186 . . . . . . . 8 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥𝐹𝑧)
2214, 21bitrdi 196 . . . . . . 7 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2322pm5.32i 454 . . . . . 6 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
24233exbii 1607 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
2513, 24bitr3i 186 . . . 4 ((∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧))
2612, 25bitrdi 196 . . 3 (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)))
2726abbi2dv 2296 . 2 (Rel dom 𝐹 → tpos 𝐹 = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)})
28 df-oprab 5872 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥𝐹𝑧)}
2927, 28eqtr4di 2228 1 (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  {cab 2163  Vcvv 2737  wss 3129  cop 3594   class class class wbr 4000   × cxp 4620  ccnv 4621  dom cdm 4622  Rel wrel 4627  {coprab 5869  tpos ctpos 6238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-oprab 5872  df-tpos 6239
This theorem is referenced by:  tposoprab  6274
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