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Theorem prter3 37752
Description: For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem18.1 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Assertion
Ref Expression
prter3 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → = 𝑆)
Distinct variable group:   𝑥,𝑢,𝑦,𝐴
Allowed substitution hints:   (𝑥,𝑦,𝑢)   𝑆(𝑥,𝑦,𝑢)

Proof of Theorem prter3
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 errel 8712 . . 3 (𝑆 Er 𝐴 → Rel 𝑆)
21adantr 482 . 2 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → Rel 𝑆)
3 prtlem18.1 . . . 4 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
43relopabiv 5821 . . 3 Rel
53prtlem13 37738 . . . . . 6 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
6 simpll 766 . . . . . . . . . . . . 13 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → 𝑆 Er 𝐴)
7 simprl 770 . . . . . . . . . . . . . . 15 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → 𝑣𝐴)
8 ne0i 4335 . . . . . . . . . . . . . . . 16 (𝑧𝑣𝑣 ≠ ∅)
98ad2antll 728 . . . . . . . . . . . . . . 15 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → 𝑣 ≠ ∅)
10 eldifsn 4791 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝐴 ∖ {∅}) ↔ (𝑣𝐴𝑣 ≠ ∅))
117, 9, 10sylanbrc 584 . . . . . . . . . . . . . 14 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → 𝑣 ∈ (𝐴 ∖ {∅}))
12 simplr 768 . . . . . . . . . . . . . 14 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → ( 𝐴 / 𝑆) = (𝐴 ∖ {∅}))
1311, 12eleqtrrd 2837 . . . . . . . . . . . . 13 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → 𝑣 ∈ ( 𝐴 / 𝑆))
14 simprr 772 . . . . . . . . . . . . 13 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → 𝑧𝑣)
15 qsel 8790 . . . . . . . . . . . . 13 ((𝑆 Er 𝐴𝑣 ∈ ( 𝐴 / 𝑆) ∧ 𝑧𝑣) → 𝑣 = [𝑧]𝑆)
166, 13, 14, 15syl3anc 1372 . . . . . . . . . . . 12 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → 𝑣 = [𝑧]𝑆)
1716eleq2d 2820 . . . . . . . . . . 11 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → (𝑤𝑣𝑤 ∈ [𝑧]𝑆))
18 vex 3479 . . . . . . . . . . . 12 𝑤 ∈ V
19 vex 3479 . . . . . . . . . . . 12 𝑧 ∈ V
2018, 19elec 8747 . . . . . . . . . . 11 (𝑤 ∈ [𝑧]𝑆𝑧𝑆𝑤)
2117, 20bitrdi 287 . . . . . . . . . 10 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → (𝑤𝑣𝑧𝑆𝑤))
2221anassrs 469 . . . . . . . . 9 ((((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑣𝐴) ∧ 𝑧𝑣) → (𝑤𝑣𝑧𝑆𝑤))
2322pm5.32da 580 . . . . . . . 8 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑣𝐴) → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑣𝑧𝑆𝑤)))
2423rexbidva 3177 . . . . . . 7 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ ∃𝑣𝐴 (𝑧𝑣𝑧𝑆𝑤)))
25 simpll 766 . . . . . . . . . . . 12 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑆 Er 𝐴)
26 simpr 486 . . . . . . . . . . . 12 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑧𝑆𝑤)
2725, 26ercl 8714 . . . . . . . . . . 11 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑧 𝐴)
28 eluni2 4913 . . . . . . . . . . 11 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
2927, 28sylib 217 . . . . . . . . . 10 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → ∃𝑣𝐴 𝑧𝑣)
3029ex 414 . . . . . . . . 9 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 → ∃𝑣𝐴 𝑧𝑣))
3130pm4.71rd 564 . . . . . . . 8 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 ↔ (∃𝑣𝐴 𝑧𝑣𝑧𝑆𝑤)))
32 r19.41v 3189 . . . . . . . 8 (∃𝑣𝐴 (𝑧𝑣𝑧𝑆𝑤) ↔ (∃𝑣𝐴 𝑧𝑣𝑧𝑆𝑤))
3331, 32bitr4di 289 . . . . . . 7 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑧𝑆𝑤)))
3424, 33bitr4d 282 . . . . . 6 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ 𝑧𝑆𝑤))
355, 34bitrid 283 . . . . 5 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧 𝑤𝑧𝑆𝑤))
3635adantl 483 . . . 4 (((Rel ∧ Rel 𝑆) ∧ (𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → (𝑧 𝑤𝑧𝑆𝑤))
3736eqbrrdv2 37733 . . 3 (((Rel ∧ Rel 𝑆) ∧ (𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → = 𝑆)
384, 37mpanl1 699 . 2 ((Rel 𝑆 ∧ (𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → = 𝑆)
392, 38mpancom 687 1 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2941  wrex 3071  cdif 3946  c0 4323  {csn 4629   cuni 4909   class class class wbr 5149  {copab 5211  Rel wrel 5682   Er wer 8700  [cec 8701   / cqs 8702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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-rn 5688  df-res 5689  df-ima 5690  df-er 8703  df-ec 8705  df-qs 8709
This theorem is referenced by: (None)
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