Theorem dftpos3 6159

Description: Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4547. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
dftpos3	⊢ (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧})

Distinct variable group:   𝑥,𝑦,𝑧,𝐹

Proof of Theorem dftpos3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 4917	. . . . . . . . . 10 ⊢ Rel ◡dom 𝐹
2		dmtpos 6153	. . . . . . . . . . 11 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
3	2	releqd 4623	. . . . . . . . . 10 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ↔ Rel ◡dom 𝐹))
4	1, 3	mpbiri 167	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹)
5		reltpos 6147	. . . . . . . . 9 ⊢ Rel tpos 𝐹
6	4, 5	jctil 310	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹))
7		relrelss 5065	. . . . . . . 8 ⊢ ((Rel tpos 𝐹 ∧ Rel dom tpos 𝐹) ↔ tpos 𝐹 ⊆ ((V × V) × V))
8	6, 7	sylib 121	. . . . . . 7 ⊢ (Rel dom 𝐹 → tpos 𝐹 ⊆ ((V × V) × V))
9	8	sseld 3096	. . . . . 6 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → 𝑤 ∈ ((V × V) × V)))
10		elvvv 4602	. . . . . 6 ⊢ (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
11	9, 10	syl6ib 160	. . . . 5 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
12	11	pm4.71rd 391	. . . 4 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹)))
13		19.41vvv 1876	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹))
14		eleq1 2202	. . . . . . . 8 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹))
15		df-br 3930	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹)
16		vex 2689	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17		vex 2689	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
18		vex 2689	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
19		brtposg 6151	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
20	16, 17, 18, 19	mp3an 1315	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)
21	15, 20	bitr3i 185	. . . . . . . 8 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)
22	14, 21	syl6bb 195	. . . . . . 7 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
23	22	pm5.32i 449	. . . . . 6 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧))
24	23	3exbii 1586	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧))
25	13, 24	bitr3i 185	. . . 4 ⊢ ((∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧))
26	12, 25	syl6bb 195	. . 3 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)))
27	26	abbi2dv 2258	. 2 ⊢ (Rel dom 𝐹 → tpos 𝐹 = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)})
28		df-oprab 5778	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)}
29	27, 28	syl6eqr 2190	1 ⊢ (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧})

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2125  Vcvv 2686   ⊆ wss 3071  ⟨cop 3530   class class class wbr 3929   × cxp 4537  ^◡ccnv 4538  dom cdm 4539  Rel wrel 4544  {coprab 5775  tpos ctpos 6141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-oprab 5778  df-tpos 6142

This theorem is referenced by:  tposoprab  6177