Theorem dftpos3 6428

Description: Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4733. (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 5114	. . . . . . . . . 10 ⊢ Rel ◡dom 𝐹
2		dmtpos 6422	. . . . . . . . . . 11 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
3	2	releqd 4810	. . . . . . . . . 10 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ↔ Rel ◡dom 𝐹))
4	1, 3	mpbiri 168	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹)
5		reltpos 6416	. . . . . . . . 9 ⊢ Rel tpos 𝐹
6	4, 5	jctil 312	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹))
7		relrelss 5263	. . . . . . . 8 ⊢ ((Rel tpos 𝐹 ∧ Rel dom tpos 𝐹) ↔ tpos 𝐹 ⊆ ((V × V) × V))
8	6, 7	sylib 122	. . . . . . 7 ⊢ (Rel dom 𝐹 → tpos 𝐹 ⊆ ((V × V) × V))
9	8	sseld 3226	. . . . . 6 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → 𝑤 ∈ ((V × V) × V)))
10		elvvv 4789	. . . . . 6 ⊢ (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
11	9, 10	imbitrdi 161	. . . . 5 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
12	11	pm4.71rd 394	. . . 4 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹)))
13		19.41vvv 1953	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹))
14		eleq1 2294	. . . . . . . 8 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹))
15		df-br 4089	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹)
16		vex 2805	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17		vex 2805	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
18		vex 2805	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
19		brtposg 6420	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
20	16, 17, 18, 19	mp3an 1373	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)
21	15, 20	bitr3i 186	. . . . . . . 8 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)
22	14, 21	bitrdi 196	. . . . . . 7 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
23	22	pm5.32i 454	. . . . . 6 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧))
24	23	3exbii 1655	. . . . 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 186	. . . 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 196	. . 3 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)))
27	26	abbi2dv 2350	. 2 ⊢ (Rel dom 𝐹 → tpos 𝐹 = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)})
28		df-oprab 6022	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)}
29	27, 28	eqtr4di 2282	1 ⊢ (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {cab 2217  Vcvv 2802   ⊆ wss 3200  ⟨cop 3672   class class class wbr 4088   × cxp 4723  ^◡ccnv 4724  dom cdm 4725  Rel wrel 4730  {coprab 6019  tpos ctpos 6410

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-oprab 6022  df-tpos 6411

This theorem is referenced by:  tposoprab  6446