Theorem prter1 34164

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	relopabi 5245	. . 3 ⊢ Rel ∼
3	2	a1i 11	. 2 ⊢ (Prt 𝐴 → Rel ∼ )
4	1	prtlem16 34154	. . 3 ⊢ dom ∼ = ∪ 𝐴
5	4	a1i 11	. 2 ⊢ (Prt 𝐴 → dom ∼ = ∪ 𝐴)
6		prtlem15 34160	. . . . . 6 ⊢ (Prt 𝐴 → (∃𝑣 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ∧ (𝑤 ∈ 𝑞 ∧ 𝑝 ∈ 𝑞)) → ∃𝑟 ∈ 𝐴 (𝑧 ∈ 𝑟 ∧ 𝑝 ∈ 𝑟)))
7	1	prtlem13 34153	. . . . . . . 8 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
8	1	prtlem13 34153	. . . . . . . 8 ⊢ (𝑤 ∼ 𝑝 ↔ ∃𝑞 ∈ 𝐴 (𝑤 ∈ 𝑞 ∧ 𝑝 ∈ 𝑞))
9	7, 8	anbi12i 733	. . . . . . 7 ⊢ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) ↔ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ∧ ∃𝑞 ∈ 𝐴 (𝑤 ∈ 𝑞 ∧ 𝑝 ∈ 𝑞)))
10		reeanv 3107	. . . . . . 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 267	. . . . . 6 ⊢ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) ↔ ∃𝑣 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ∧ (𝑤 ∈ 𝑞 ∧ 𝑝 ∈ 𝑞)))
12	1	prtlem13 34153	. . . . . 6 ⊢ (𝑧 ∼ 𝑝 ↔ ∃𝑟 ∈ 𝐴 (𝑧 ∈ 𝑟 ∧ 𝑝 ∈ 𝑟))
13	6, 11, 12	3imtr4g 285	. . . . 5 ⊢ (Prt 𝐴 → ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) → 𝑧 ∼ 𝑝))
14		pm3.22 465	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → (𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣))
15	14	reximi 3011	. . . . . 6 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → ∃𝑣 ∈ 𝐴 (𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣))
16	1	prtlem13 34153	. . . . . 6 ⊢ (𝑤 ∼ 𝑧 ↔ ∃𝑣 ∈ 𝐴 (𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣))
17	15, 7, 16	3imtr4i 281	. . . . 5 ⊢ (𝑧 ∼ 𝑤 → 𝑤 ∼ 𝑧)
18	13, 17	jctil 560	. . . 4 ⊢ (Prt 𝐴 → ((𝑧 ∼ 𝑤 → 𝑤 ∼ 𝑧) ∧ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) → 𝑧 ∼ 𝑝)))
19	18	alrimivv 1856	. . 3 ⊢ (Prt 𝐴 → ∀𝑤∀𝑝((𝑧 ∼ 𝑤 → 𝑤 ∼ 𝑧) ∧ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) → 𝑧 ∼ 𝑝)))
20	19	alrimiv 1855	. 2 ⊢ (Prt 𝐴 → ∀𝑧∀𝑤∀𝑝((𝑧 ∼ 𝑤 → 𝑤 ∼ 𝑧) ∧ ((𝑧 ∼ 𝑤 ∧ 𝑤 ∼ 𝑝) → 𝑧 ∼ 𝑝)))
21		dfer2 7743	. 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 1246	1 ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wrex 2913  ∪ cuni 4436   class class class wbr 4653  {copab 4712  dom cdm 5114  Rel wrel 5119   Er wer 7739  Prt wprt 34156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-er 7742  df-prt 34157

This theorem is referenced by:  prtex  34165