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Theorem prter3 36177
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 8285 . . 3 (𝑆 Er 𝐴 → Rel 𝑆)
21adantr 484 . 2 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → Rel 𝑆)
3 prtlem18.1 . . . 4 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
43relopabi 5662 . . 3 Rel
53prtlem13 36163 . . . . . 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 4253 . . . . . . . . . . . . . . . 16 (𝑧𝑣𝑣 ≠ ∅)
98ad2antll 728 . . . . . . . . . . . . . . 15 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → 𝑣 ≠ ∅)
10 eldifsn 4683 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝐴 ∖ {∅}) ↔ (𝑣𝐴𝑣 ≠ ∅))
117, 9, 10sylanbrc 586 . . . . . . . . . . . . . 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 2896 . . . . . . . . . . . . 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 8363 . . . . . . . . . . . . 13 ((𝑆 Er 𝐴𝑣 ∈ ( 𝐴 / 𝑆) ∧ 𝑧𝑣) → 𝑣 = [𝑧]𝑆)
166, 13, 14, 15syl3anc 1368 . . . . . . . . . . . 12 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → 𝑣 = [𝑧]𝑆)
1716eleq2d 2878 . . . . . . . . . . 11 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → (𝑤𝑣𝑤 ∈ [𝑧]𝑆))
18 vex 3447 . . . . . . . . . . . 12 𝑤 ∈ V
19 vex 3447 . . . . . . . . . . . 12 𝑧 ∈ V
2018, 19elec 8320 . . . . . . . . . . 11 (𝑤 ∈ [𝑧]𝑆𝑧𝑆𝑤)
2117, 20syl6bb 290 . . . . . . . . . 10 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣𝐴𝑧𝑣)) → (𝑤𝑣𝑧𝑆𝑤))
2221anassrs 471 . . . . . . . . 9 ((((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑣𝐴) ∧ 𝑧𝑣) → (𝑤𝑣𝑧𝑆𝑤))
2322pm5.32da 582 . . . . . . . 8 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑣𝐴) → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑣𝑧𝑆𝑤)))
2423rexbidva 3258 . . . . . . 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 488 . . . . . . . . . . . 12 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑧𝑆𝑤)
2725, 26ercl 8287 . . . . . . . . . . 11 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑧 𝐴)
28 eluni2 4807 . . . . . . . . . . 11 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
2927, 28sylib 221 . . . . . . . . . 10 (((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → ∃𝑣𝐴 𝑧𝑣)
3029ex 416 . . . . . . . . 9 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 → ∃𝑣𝐴 𝑧𝑣))
3130pm4.71rd 566 . . . . . . . 8 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 ↔ (∃𝑣𝐴 𝑧𝑣𝑧𝑆𝑤)))
32 r19.41v 3303 . . . . . . . 8 (∃𝑣𝐴 (𝑧𝑣𝑧𝑆𝑤) ↔ (∃𝑣𝐴 𝑧𝑣𝑧𝑆𝑤))
3331, 32syl6bbr 292 . . . . . . 7 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑧𝑆𝑤)))
3424, 33bitr4d 285 . . . . . 6 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ 𝑧𝑆𝑤))
355, 34syl5bb 286 . . . . 5 ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧 𝑤𝑧𝑆𝑤))
3635adantl 485 . . . 4 (((Rel ∧ Rel 𝑆) ∧ (𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → (𝑧 𝑤𝑧𝑆𝑤))
3736eqbrrdv2 36158 . . 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 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wrex 3110  cdif 3881  c0 4246  {csn 4528   cuni 4803   class class class wbr 5033  {copab 5095  Rel wrel 5528   Er wer 8273  [cec 8274   / cqs 8275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-er 8276  df-ec 8278  df-qs 8282
This theorem is referenced by: (None)
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