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Theorem prter1 34687
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 5384 . . 3 Rel
32a1i 11 . 2 (Prt 𝐴 → Rel )
41prtlem16 34677 . . 3 dom = 𝐴
54a1i 11 . 2 (Prt 𝐴 → dom = 𝐴)
6 prtlem15 34683 . . . . . 6 (Prt 𝐴 → (∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)) → ∃𝑟𝐴 (𝑧𝑟𝑝𝑟)))
71prtlem13 34676 . . . . . . . 8 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
81prtlem13 34676 . . . . . . . 8 (𝑤 𝑝 ↔ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞))
97, 8anbi12i 612 . . . . . . 7 ((𝑧 𝑤𝑤 𝑝) ↔ (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ∧ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞)))
10 reeanv 3255 . . . . . . 7 (∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)) ↔ (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ∧ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞)))
119, 10bitr4i 267 . . . . . 6 ((𝑧 𝑤𝑤 𝑝) ↔ ∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)))
121prtlem13 34676 . . . . . 6 (𝑧 𝑝 ↔ ∃𝑟𝐴 (𝑧𝑟𝑝𝑟))
136, 11, 123imtr4g 285 . . . . 5 (Prt 𝐴 → ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝))
14 pm3.22 449 . . . . . . 7 ((𝑧𝑣𝑤𝑣) → (𝑤𝑣𝑧𝑣))
1514reximi 3159 . . . . . 6 (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) → ∃𝑣𝐴 (𝑤𝑣𝑧𝑣))
161prtlem13 34676 . . . . . 6 (𝑤 𝑧 ↔ ∃𝑣𝐴 (𝑤𝑣𝑧𝑣))
1715, 7, 163imtr4i 281 . . . . 5 (𝑧 𝑤𝑤 𝑧)
1813, 17jctil 509 . . . 4 (Prt 𝐴 → ((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
1918alrimivv 2008 . . 3 (Prt 𝐴 → ∀𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
2019alrimiv 2007 . 2 (Prt 𝐴 → ∀𝑧𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
21 dfer2 7897 . 2 ( Er 𝐴 ↔ (Rel ∧ dom = 𝐴 ∧ ∀𝑧𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝))))
223, 5, 20, 21syl3anbrc 1428 1 (Prt 𝐴 Er 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629   = wceq 1631  wrex 3062   cuni 4574   class class class wbr 4786  {copab 4846  dom cdm 5249  Rel wrel 5254   Er wer 7893  Prt wprt 34679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-er 7896  df-prt 34680
This theorem is referenced by:  prtex  34688
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