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Theorem prter1 33683
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 5215 . . 3 Rel
32a1i 11 . 2 (Prt 𝐴 → Rel )
41prtlem16 33673 . . 3 dom = 𝐴
54a1i 11 . 2 (Prt 𝐴 → dom = 𝐴)
6 prtlem15 33679 . . . . . 6 (Prt 𝐴 → (∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)) → ∃𝑟𝐴 (𝑧𝑟𝑝𝑟)))
71prtlem13 33672 . . . . . . . 8 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
81prtlem13 33672 . . . . . . . 8 (𝑤 𝑝 ↔ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞))
97, 8anbi12i 732 . . . . . . 7 ((𝑧 𝑤𝑤 𝑝) ↔ (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ∧ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞)))
10 reeanv 3101 . . . . . . 7 (∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)) ↔ (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ∧ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞)))
119, 10bitr4i 267 . . . . . 6 ((𝑧 𝑤𝑤 𝑝) ↔ ∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)))
121prtlem13 33672 . . . . . 6 (𝑧 𝑝 ↔ ∃𝑟𝐴 (𝑧𝑟𝑝𝑟))
136, 11, 123imtr4g 285 . . . . 5 (Prt 𝐴 → ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝))
14 pm3.22 465 . . . . . . 7 ((𝑧𝑣𝑤𝑣) → (𝑤𝑣𝑧𝑣))
1514reximi 3007 . . . . . 6 (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) → ∃𝑣𝐴 (𝑤𝑣𝑧𝑣))
161prtlem13 33672 . . . . . 6 (𝑤 𝑧 ↔ ∃𝑣𝐴 (𝑤𝑣𝑧𝑣))
1715, 7, 163imtr4i 281 . . . . 5 (𝑧 𝑤𝑤 𝑧)
1813, 17jctil 559 . . . 4 (Prt 𝐴 → ((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
1918alrimivv 1853 . . 3 (Prt 𝐴 → ∀𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
2019alrimiv 1852 . 2 (Prt 𝐴 → ∀𝑧𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
21 dfer2 7703 . 2 ( Er 𝐴 ↔ (Rel ∧ dom = 𝐴 ∧ ∀𝑧𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝))))
223, 5, 20, 21syl3anbrc 1244 1 (Prt 𝐴 Er 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wrex 2909   cuni 4409   class class class wbr 4623  {copab 4682  dom cdm 5084  Rel wrel 5089   Er wer 7699  Prt wprt 33675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-er 7702  df-prt 33676
This theorem is referenced by:  prtex  33684
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