Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prter1 Structured version   Visualization version   GIF version

Theorem prter1 36002
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 5687 . . 3 Rel
32a1i 11 . 2 (Prt 𝐴 → Rel )
41prtlem16 35992 . . 3 dom = 𝐴
54a1i 11 . 2 (Prt 𝐴 → dom = 𝐴)
6 prtlem15 35998 . . . . . 6 (Prt 𝐴 → (∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)) → ∃𝑟𝐴 (𝑧𝑟𝑝𝑟)))
71prtlem13 35991 . . . . . . . 8 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
81prtlem13 35991 . . . . . . . 8 (𝑤 𝑝 ↔ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞))
97, 8anbi12i 628 . . . . . . 7 ((𝑧 𝑤𝑤 𝑝) ↔ (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ∧ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞)))
10 reeanv 3366 . . . . . . 7 (∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)) ↔ (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ∧ ∃𝑞𝐴 (𝑤𝑞𝑝𝑞)))
119, 10bitr4i 280 . . . . . 6 ((𝑧 𝑤𝑤 𝑝) ↔ ∃𝑣𝐴𝑞𝐴 ((𝑧𝑣𝑤𝑣) ∧ (𝑤𝑞𝑝𝑞)))
121prtlem13 35991 . . . . . 6 (𝑧 𝑝 ↔ ∃𝑟𝐴 (𝑧𝑟𝑝𝑟))
136, 11, 123imtr4g 298 . . . . 5 (Prt 𝐴 → ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝))
14 pm3.22 462 . . . . . . 7 ((𝑧𝑣𝑤𝑣) → (𝑤𝑣𝑧𝑣))
1514reximi 3241 . . . . . 6 (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) → ∃𝑣𝐴 (𝑤𝑣𝑧𝑣))
161prtlem13 35991 . . . . . 6 (𝑤 𝑧 ↔ ∃𝑣𝐴 (𝑤𝑣𝑧𝑣))
1715, 7, 163imtr4i 294 . . . . 5 (𝑧 𝑤𝑤 𝑧)
1813, 17jctil 522 . . . 4 (Prt 𝐴 → ((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
1918alrimivv 1922 . . 3 (Prt 𝐴 → ∀𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
2019alrimiv 1921 . 2 (Prt 𝐴 → ∀𝑧𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝)))
21 dfer2 8282 . 2 ( Er 𝐴 ↔ (Rel ∧ dom = 𝐴 ∧ ∀𝑧𝑤𝑝((𝑧 𝑤𝑤 𝑧) ∧ ((𝑧 𝑤𝑤 𝑝) → 𝑧 𝑝))))
223, 5, 20, 21syl3anbrc 1337 1 (Prt 𝐴 Er 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wal 1528   = wceq 1530  wrex 3137   cuni 4830   class class class wbr 5057  {copab 5119  dom cdm 5548  Rel wrel 5553   Er wer 8278  Prt wprt 35994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-er 8281  df-prt 35995
This theorem is referenced by:  prtex  36003
  Copyright terms: Public domain W3C validator