Theorem prter1 37414

Description: Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem18.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}

Assertion

Ref	Expression
prter1	⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴)

Distinct variable group:   𝑥,𝑢,𝑦,𝐴

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑢)

Proof of Theorem prter1

Dummy variables 𝑞 𝑝 𝑟 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prtlem18.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
2	1	relopabiv 5781	. . 3 ⊢ Rel ∼
3	2	a1i 11	. 2 ⊢ (Prt 𝐴 → Rel ∼ )
4	1	prtlem16 37404	. . 3 ⊢ dom ∼ = ∪ 𝐴
5	4	a1i 11	. 2 ⊢ (Prt 𝐴 → dom ∼ = ∪ 𝐴)
6		prtlem15 37410	. . . . . 6 ⊢ (Prt 𝐴 → (∃𝑣 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ∧ (𝑤 ∈ 𝑞 ∧ 𝑝 ∈ 𝑞)) → ∃𝑟 ∈ 𝐴 (𝑧 ∈ 𝑟 ∧ 𝑝 ∈ 𝑟)))
7	1	prtlem13 37403	. . . . . . . 8 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
8	1	prtlem13 37403	. . . . . . . 8 ⊢ (𝑤 ∼ 𝑝 ↔ ∃𝑞 ∈ 𝐴 (𝑤 ∈ 𝑞 ∧ 𝑝 ∈ 𝑞))
9	7, 8	anbi12i 627	. . . . . . 7 ⊢ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) ↔ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ∧ ∃𝑞 ∈ 𝐴 (𝑤 ∈ 𝑞 ∧ 𝑝 ∈ 𝑞)))
10		reeanv 3215	. . . . . . 7 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ∧ (𝑤 ∈ 𝑞 ∧ 𝑝 ∈ 𝑞)) ↔ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ∧ ∃𝑞 ∈ 𝐴 (𝑤 ∈ 𝑞 ∧ 𝑝 ∈ 𝑞)))
11	9, 10	bitr4i 277	. . . . . 6 ⊢ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) ↔ ∃𝑣 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ∧ (𝑤 ∈ 𝑞 ∧ 𝑝 ∈ 𝑞)))
12	1	prtlem13 37403	. . . . . 6 ⊢ (𝑧 ∼ 𝑝 ↔ ∃𝑟 ∈ 𝐴 (𝑧 ∈ 𝑟 ∧ 𝑝 ∈ 𝑟))
13	6, 11, 12	3imtr4g 295	. . . . 5 ⊢ (Prt 𝐴 → ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) → 𝑧 ∼ 𝑝))
14		pm3.22 460	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → (𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣))
15	14	reximi 3083	. . . . . 6 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → ∃𝑣 ∈ 𝐴 (𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣))
16	1	prtlem13 37403	. . . . . 6 ⊢ (𝑤 ∼ 𝑧 ↔ ∃𝑣 ∈ 𝐴 (𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣))
17	15, 7, 16	3imtr4i 291	. . . . 5 ⊢ (𝑧 ∼ 𝑤 → 𝑤 ∼ 𝑧)
18	13, 17	jctil 520	. . . 4 ⊢ (Prt 𝐴 → ((𝑧 ∼ 𝑤 → 𝑤 ∼ 𝑧) ∧ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) → 𝑧 ∼ 𝑝)))
19	18	alrimivv 1931	. . 3 ⊢ (Prt 𝐴 → ∀𝑤∀𝑝((𝑧 ∼ 𝑤 → 𝑤 ∼ 𝑧) ∧ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) → 𝑧 ∼ 𝑝)))
20	19	alrimiv 1930	. 2 ⊢ (Prt 𝐴 → ∀𝑧∀𝑤∀𝑝((𝑧 ∼ 𝑤 → 𝑤 ∼ 𝑧) ∧ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) → 𝑧 ∼ 𝑝)))
21		dfer2 8656	. 2 ⊢ ( ∼ Er ∪ 𝐴 ↔ (Rel ∼ ∧ dom ∼ = ∪ 𝐴 ∧ ∀𝑧∀𝑤∀𝑝((𝑧 ∼ 𝑤 → 𝑤 ∼ 𝑧) ∧ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) → 𝑧 ∼ 𝑝))))
22	3, 5, 20, 21	syl3anbrc 1343	1 ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1539   = wceq 1541  ∃wrex 3069  ∪ cuni 4870   class class class wbr 5110  {copab 5172  dom cdm 5638  Rel wrel 5643   Er wer 8652  Prt wprt 37406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-er 8655  df-prt 37407

This theorem is referenced by:  prtex  37415