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Theorem prter1 38054
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)}
21relopabiv 5821 . . 3 Rel
32a1i 11 . 2 (Prt 𝐴 → Rel )
41prtlem16 38044 . . 3 dom = 𝐴
54a1i 11 . 2 (Prt 𝐴 → dom = 𝐴)
6 prtlem15 38050 . . . . . 6 (Prt 𝐴 → (∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)) → ∃𝑟𝐴 (𝑧𝑟𝑝𝑟)))
71prtlem13 38043 . . . . . . . 8 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
81prtlem13 38043 . . . . . . . 8 (𝑤 𝑝 ↔ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞))
97, 8anbi12i 625 . . . . . . 7 ((𝑧 𝑤𝑤 𝑝) ↔ (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ∧ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞)))
10 reeanv 3224 . . . . . . 7 (∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)) ↔ (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ∧ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞)))
119, 10bitr4i 277 . . . . . 6 ((𝑧 𝑤𝑤 𝑝) ↔ ∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)))
121prtlem13 38043 . . . . . 6 (𝑧 𝑝 ↔ ∃𝑟𝐴 (𝑧𝑟𝑝𝑟))
136, 11, 123imtr4g 295 . . . . 5 (Prt 𝐴 → ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝))
14 pm3.22 458 . . . . . . 7 ((𝑧𝑣𝑤𝑣) → (𝑤𝑣𝑧𝑣))
1514reximi 3082 . . . . . 6 (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) → ∃𝑣𝐴 (𝑤𝑣𝑧𝑣))
161prtlem13 38043 . . . . . 6 (𝑤 𝑧 ↔ ∃𝑣𝐴 (𝑤𝑣𝑧𝑣))
1715, 7, 163imtr4i 291 . . . . 5 (𝑧 𝑤𝑤 𝑧)
1813, 17jctil 518 . . . 4 (Prt 𝐴 → ((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
1918alrimivv 1929 . . 3 (Prt 𝐴 → ∀𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
2019alrimiv 1928 . 2 (Prt 𝐴 → ∀𝑧𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
21 dfer2 8708 . 2 ( Er 𝐴 ↔ (Rel ∧ dom = 𝐴 ∧ ∀𝑧𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝))))
223, 5, 20, 21syl3anbrc 1341 1 (Prt 𝐴 Er 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1537   = wceq 1539  wrex 3068   cuni 4909   class class class wbr 5149  {copab 5211  dom cdm 5677  Rel wrel 5682   Er wer 8704  Prt wprt 38046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-er 8707  df-prt 38047
This theorem is referenced by:  prtex  38055
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