Theorem dftpos3 8241

Description: Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 5671. (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 6108	. . . . . . . . . 10 ⊢ Rel ◡dom 𝐹
2		dmtpos 8235	. . . . . . . . . . 11 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
3	2	releqd 5767	. . . . . . . . . 10 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ↔ Rel ◡dom 𝐹))
4	1, 3	mpbiri 261	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹)
5		reltpos 8228	. . . . . . . . 9 ⊢ Rel tpos 𝐹
6	4, 5	jctil 528	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹))
7		relrelss 6276	. . . . . . . 8 ⊢ ((Rel tpos 𝐹 ∧ Rel dom tpos 𝐹) ↔ tpos 𝐹 ⊆ ((V × V) × V))
8	6, 7	sylib 221	. . . . . . 7 ⊢ (Rel dom 𝐹 → tpos 𝐹 ⊆ ((V × V) × V))
9	8	sseld 3937	. . . . . 6 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → 𝑤 ∈ ((V × V) × V)))
10		elvvv 5739	. . . . . 6 ⊢ (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
11	9, 10	imbitrdi 254	. . . . 5 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 → ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
12	11	pm4.71rd 571	. . . 4 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹)))
13		19.41vvv 1981	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹))
14		eleq1 2851	. . . . . . . 8 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹))
15		df-br 5111	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹)
16		brtpos 8232	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
17	16	elv 3460	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)
18	15, 17	bitr3i 280	. . . . . . . 8 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)
19	14, 18	bitrdi 290	. . . . . . 7 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
20	19	pm5.32i 584	. . . . . 6 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧))
21	20	3exbii 1880	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧))
22	13, 21	bitr3i 280	. . . 4 ⊢ ((∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧))
23	12, 22	bitrdi 290	. . 3 ⊢ (Rel dom 𝐹 → (𝑤 ∈ tpos 𝐹 ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)))
24	23	eqabdv 2896	. 2 ⊢ (Rel dom 𝐹 → tpos 𝐹 = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)})
25		df-oprab 7416	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)}
26	24, 25	eqtr4di 2816	1 ⊢ (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  Vcvv 3455   ⊆ wss 3906  ⟨cop 4596   class class class wbr 5110   × cxp 5661  ^◡ccnv 5662  dom cdm 5663  Rel wrel 5668  {coprab 7413  tpos ctpos 8222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-oprab 7416  df-tpos 8223

This theorem is referenced by:  tposoprab  8259