Theorem dftpos3 6241

Description: Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4619. (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 4989	. . . . . . . . . 10 ⊢ Rel ◡dom 𝐹
2		dmtpos 6235	. . . . . . . . . . 11 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
3	2	releqd 4695	. . . . . . . . . 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 6229	. . . . . . . . 9 ⊢ Rel tpos 𝐹
6	4, 5	jctil 310	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹))
7		relrelss 5137	. . . . . . . 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 3146	. . . . . 6 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → 𝑤 ∈ ((V × V) × V)))
10		elvvv 4674	. . . . . 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 392	. . . 4 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹)))
13		19.41vvv 1897	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹))
14		eleq1 2233	. . . . . . . 8 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹))
15		df-br 3990	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹)
16		vex 2733	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17		vex 2733	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
18		vex 2733	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
19		brtposg 6233	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
20	16, 17, 18, 19	mp3an 1332	. . . . . . . . 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	bitrdi 195	. . . . . . 7 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
23	22	pm5.32i 451	. . . . . 6 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧))
24	23	3exbii 1600	. . . . 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	bitrdi 195	. . 3 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)))
27	26	abbi2dv 2289	. 2 ⊢ (Rel dom 𝐹 → tpos 𝐹 = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)})
28		df-oprab 5857	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)}
29	27, 28	eqtr4di 2221	1 ⊢ (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧})

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  Vcvv 2730   ⊆ wss 3121  ⟨cop 3586   class class class wbr 3989   × cxp 4609  ^◡ccnv 4610  dom cdm 4611  Rel wrel 4616  {coprab 5854  tpos ctpos 6223

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-oprab 5857  df-tpos 6224

This theorem is referenced by:  tposoprab  6259