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Theorem prter1 34900
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⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
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.
StepHypRef Expression
1 prtlem18.1 . . . 4 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
21relopabi 5449 . . 3 Rel
32a1i 11 . 2 (Prt 𝐴 → Rel )
41prtlem16 34890 . . 3 dom = 𝐴
54a1i 11 . 2 (Prt 𝐴 → dom = 𝐴)
6 prtlem15 34896 . . . . . 6 (Prt 𝐴 → (∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)) → ∃𝑟𝐴 (𝑧𝑟𝑝𝑟)))
71prtlem13 34889 . . . . . . . 8 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
81prtlem13 34889 . . . . . . . 8 (𝑤 𝑝 ↔ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞))
97, 8anbi12i 621 . . . . . . 7 ((𝑧 𝑤𝑤 𝑝) ↔ (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ∧ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞)))
10 reeanv 3288 . . . . . . 7 (∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)) ↔ (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ∧ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞)))
119, 10bitr4i 270 . . . . . 6 ((𝑧 𝑤𝑤 𝑝) ↔ ∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)))
121prtlem13 34889 . . . . . 6 (𝑧 𝑝 ↔ ∃𝑟𝐴 (𝑧𝑟𝑝𝑟))
136, 11, 123imtr4g 288 . . . . 5 (Prt 𝐴 → ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝))
14 pm3.22 452 . . . . . . 7 ((𝑧𝑣𝑤𝑣) → (𝑤𝑣𝑧𝑣))
1514reximi 3191 . . . . . 6 (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) → ∃𝑣𝐴 (𝑤𝑣𝑧𝑣))
161prtlem13 34889 . . . . . 6 (𝑤 𝑧 ↔ ∃𝑣𝐴 (𝑤𝑣𝑧𝑣))
1715, 7, 163imtr4i 284 . . . . 5 (𝑧 𝑤𝑤 𝑧)
1813, 17jctil 516 . . . 4 (Prt 𝐴 → ((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
1918alrimivv 2024 . . 3 (Prt 𝐴 → ∀𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
2019alrimiv 2023 . 2 (Prt 𝐴 → ∀𝑧𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
21 dfer2 7983 . 2 ( Er 𝐴 ↔ (Rel ∧ dom = 𝐴 ∧ ∀𝑧𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝))))
223, 5, 20, 21syl3anbrc 1444 1 (Prt 𝐴 Er 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wal 1651   = wceq 1653  wrex 3090   cuni 4628   class class class wbr 4843  {copab 4905  dom cdm 5312  Rel wrel 5317   Er wer 7979  Prt wprt 34892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-er 7982  df-prt 34893
This theorem is referenced by:  prtex  34901
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