Theorem dftpos3 5795

Description: Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4276. (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 4626	. . . . . . . . . 10 ⊢ Rel ◡dom 𝐹
2		dmtpos 5789	. . . . . . . . . . 11 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
3	2	releqd 4347	. . . . . . . . . 10 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ↔ Rel ◡dom 𝐹))
4	1, 3	mpbiri 157	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹)
5		reltpos 5783	. . . . . . . . 9 ⊢ Rel tpos 𝐹
6	4, 5	jctil 295	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹))
7		relrelss 4767	. . . . . . . 8 ⊢ ((Rel tpos 𝐹 ∧ Rel dom tpos 𝐹) ↔ tpos 𝐹 ⊆ ((V × V) × V))
8	6, 7	sylib 127	. . . . . . 7 ⊢ (Rel dom 𝐹 → tpos 𝐹 ⊆ ((V × V) × V))
9	8	sseld 2917	. . . . . 6 ⊢ (Rel dom 𝐹 → (w ∈ tpos 𝐹 → w ∈ ((V × V) × V)))
10		elvvv 4326	. . . . . 6 ⊢ (w ∈ ((V × V) × V) ↔ ∃x∃y∃z w = ⟨⟨x, y⟩, z⟩)
11	9, 10	syl6ib 150	. . . . 5 ⊢ (Rel dom 𝐹 → (w ∈ tpos 𝐹 → ∃x∃y∃z w = ⟨⟨x, y⟩, z⟩))
12	11	pm4.71rd 374	. . . 4 ⊢ (Rel dom 𝐹 → (w ∈ tpos 𝐹 ↔ (∃x∃y∃z w = ⟨⟨x, y⟩, z⟩ ∧ w ∈ tpos 𝐹)))
13		19.41vvv 1762	. . . . 5 ⊢ (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ w ∈ tpos 𝐹) ↔ (∃x∃y∃z w = ⟨⟨x, y⟩, z⟩ ∧ w ∈ tpos 𝐹))
14		eleq1 2078	. . . . . . . 8 ⊢ (w = ⟨⟨x, y⟩, z⟩ → (w ∈ tpos 𝐹 ↔ ⟨⟨x, y⟩, z⟩ ∈ tpos 𝐹))
15		df-br 3735	. . . . . . . . 9 ⊢ (⟨x, y⟩tpos 𝐹z ↔ ⟨⟨x, y⟩, z⟩ ∈ tpos 𝐹)
16		vex 2534	. . . . . . . . . 10 ⊢ x ∈ V
17		vex 2534	. . . . . . . . . 10 ⊢ y ∈ V
18		vex 2534	. . . . . . . . . 10 ⊢ z ∈ V
19		brtposg 5787	. . . . . . . . . 10 ⊢ ((x ∈ V ∧ y ∈ V ∧ z ∈ V) → (⟨x, y⟩tpos 𝐹z ↔ ⟨y, x⟩𝐹z))
20	16, 17, 18, 19	mp3an 1215	. . . . . . . . 9 ⊢ (⟨x, y⟩tpos 𝐹z ↔ ⟨y, x⟩𝐹z)
21	15, 20	bitr3i 175	. . . . . . . 8 ⊢ (⟨⟨x, y⟩, z⟩ ∈ tpos 𝐹 ↔ ⟨y, x⟩𝐹z)
22	14, 21	syl6bb 185	. . . . . . 7 ⊢ (w = ⟨⟨x, y⟩, z⟩ → (w ∈ tpos 𝐹 ↔ ⟨y, x⟩𝐹z))
23	22	pm5.32i 430	. . . . . 6 ⊢ ((w = ⟨⟨x, y⟩, z⟩ ∧ w ∈ tpos 𝐹) ↔ (w = ⟨⟨x, y⟩, z⟩ ∧ ⟨y, x⟩𝐹z))
24	23	3exbii 1476	. . . . 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 175	. . . 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 185	. . 3 ⊢ (Rel dom 𝐹 → (w ∈ tpos 𝐹 ↔ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ⟨y, x⟩𝐹z)))
27	26	abbi2dv 2134	. 2 ⊢ (Rel dom 𝐹 → tpos 𝐹 = {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ⟨y, x⟩𝐹z)})
28		df-oprab 5436	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ⟨y, x⟩𝐹z} = {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ⟨y, x⟩𝐹z)}
29	27, 28	syl6eqr 2068	1 ⊢ (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ⟨y, x⟩𝐹z})

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1226  ∃wex 1358   ∈ wcel 1370  {cab 2004  Vcvv 2531   ⊆ wss 2890  ⟨cop 3349   class class class wbr 3734   × cxp 4266  ^◡ccnv 4267  dom cdm 4268  Rel wrel 4273  {coprab 5433  tpos ctpos 5777

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-sbc 2738  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833  df-oprab 5436  df-tpos 5778

This theorem is referenced by:  tposoprab  5813