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Theorem prter2 36174
Description: The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.)
Hypothesis
Ref Expression
prtlem18.1 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Assertion
Ref Expression
prter2 (Prt 𝐴 → ( 𝐴 / ) = (𝐴 ∖ {∅}))
Distinct variable group:   𝑥,𝑢,𝑦,𝐴
Allowed substitution hints:   (𝑥,𝑦,𝑢)

Proof of Theorem prter2
Dummy variables 𝑝 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3212 . . . . . . . . . . 11 (∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ))
2 r19.41v 3300 . . . . . . . . . . . 12 (∃𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ) ↔ (∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
32exbii 1849 . . . . . . . . . . 11 (∃𝑧𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
41, 3bitri 278 . . . . . . . . . 10 (∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
5 df-rex 3112 . . . . . . . . . . 11 (∃𝑧𝑣 𝑝 = [𝑧] ↔ ∃𝑧(𝑧𝑣𝑝 = [𝑧] ))
65rexbii 3210 . . . . . . . . . 10 (∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] ↔ ∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ))
7 vex 3444 . . . . . . . . . . . 12 𝑝 ∈ V
87elqs 8332 . . . . . . . . . . 11 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑧 𝐴𝑝 = [𝑧] )
9 df-rex 3112 . . . . . . . . . . . 12 (∃𝑧 𝐴𝑝 = [𝑧] ↔ ∃𝑧(𝑧 𝐴𝑝 = [𝑧] ))
10 eluni2 4804 . . . . . . . . . . . . . 14 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
1110anbi1i 626 . . . . . . . . . . . . 13 ((𝑧 𝐴𝑝 = [𝑧] ) ↔ (∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
1211exbii 1849 . . . . . . . . . . . 12 (∃𝑧(𝑧 𝐴𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
139, 12bitri 278 . . . . . . . . . . 11 (∃𝑧 𝐴𝑝 = [𝑧] ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
148, 13bitri 278 . . . . . . . . . 10 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
154, 6, 143bitr4ri 307 . . . . . . . . 9 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] )
16 prtlem18.1 . . . . . . . . . . . 12 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
1716prtlem19 36171 . . . . . . . . . . 11 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))
1817ralrimivv 3155 . . . . . . . . . 10 (Prt 𝐴 → ∀𝑣𝐴𝑧𝑣 𝑣 = [𝑧] )
19 2r19.29 3290 . . . . . . . . . . 11 ((∀𝑣𝐴𝑧𝑣 𝑣 = [𝑧] ∧ ∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] ) → ∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ))
2019ex 416 . . . . . . . . . 10 (∀𝑣𝐴𝑧𝑣 𝑣 = [𝑧] → (∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] → ∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] )))
2118, 20syl 17 . . . . . . . . 9 (Prt 𝐴 → (∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] → ∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] )))
2215, 21syl5bi 245 . . . . . . . 8 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] )))
23 eqtr3 2820 . . . . . . . . . 10 ((𝑣 = [𝑧] 𝑝 = [𝑧] ) → 𝑣 = 𝑝)
2423reximi 3206 . . . . . . . . 9 (∃𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ) → ∃𝑧𝑣 𝑣 = 𝑝)
2524reximi 3206 . . . . . . . 8 (∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ) → ∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝)
2622, 25syl6 35 . . . . . . 7 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝))
27 df-rex 3112 . . . . . . . . . 10 (∃𝑧𝑣 𝑣 = 𝑝 ↔ ∃𝑧(𝑧𝑣𝑣 = 𝑝))
28 19.41v 1950 . . . . . . . . . 10 (∃𝑧(𝑧𝑣𝑣 = 𝑝) ↔ (∃𝑧 𝑧𝑣𝑣 = 𝑝))
2927, 28bitri 278 . . . . . . . . 9 (∃𝑧𝑣 𝑣 = 𝑝 ↔ (∃𝑧 𝑧𝑣𝑣 = 𝑝))
3029simprbi 500 . . . . . . . 8 (∃𝑧𝑣 𝑣 = 𝑝𝑣 = 𝑝)
3130reximi 3206 . . . . . . 7 (∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝 → ∃𝑣𝐴 𝑣 = 𝑝)
3226, 31syl6 35 . . . . . 6 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴 𝑣 = 𝑝))
33 risset 3226 . . . . . 6 (𝑝𝐴 ↔ ∃𝑣𝐴 𝑣 = 𝑝)
3432, 33syl6ibr 255 . . . . 5 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝𝐴))
3516prtlem400 36163 . . . . . 6 ¬ ∅ ∈ ( 𝐴 / )
36 nelelne 3085 . . . . . 6 (¬ ∅ ∈ ( 𝐴 / ) → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ≠ ∅))
3735, 36mp1i 13 . . . . 5 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ≠ ∅))
3834, 37jcad 516 . . . 4 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → (𝑝𝐴𝑝 ≠ ∅)))
39 eldifsn 4680 . . . 4 (𝑝 ∈ (𝐴 ∖ {∅}) ↔ (𝑝𝐴𝑝 ≠ ∅))
4038, 39syl6ibr 255 . . 3 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ∈ (𝐴 ∖ {∅})))
41 neldifsn 4685 . . . . . . 7 ¬ ∅ ∈ (𝐴 ∖ {∅})
42 n0el 4275 . . . . . . 7 (¬ ∅ ∈ (𝐴 ∖ {∅}) ↔ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧𝑝)
4341, 42mpbi 233 . . . . . 6 𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧𝑝
4443rspec 3172 . . . . 5 (𝑝 ∈ (𝐴 ∖ {∅}) → ∃𝑧 𝑧𝑝)
45 eldifi 4054 . . . . 5 (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝𝐴)
4644, 45jca 515 . . . 4 (𝑝 ∈ (𝐴 ∖ {∅}) → (∃𝑧 𝑧𝑝𝑝𝐴))
4716prtlem19 36171 . . . . . . . . 9 (Prt 𝐴 → ((𝑝𝐴𝑧𝑝) → 𝑝 = [𝑧] ))
4847ancomsd 469 . . . . . . . 8 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → 𝑝 = [𝑧] ))
49 elunii 4805 . . . . . . . 8 ((𝑧𝑝𝑝𝐴) → 𝑧 𝐴)
5048, 49jca2r 36148 . . . . . . 7 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → (𝑧 𝐴𝑝 = [𝑧] )))
51 prtlem11 36159 . . . . . . . . 9 (𝑝 ∈ V → (𝑧 𝐴 → (𝑝 = [𝑧] 𝑝 ∈ ( 𝐴 / ))))
5251elv 3446 . . . . . . . 8 (𝑧 𝐴 → (𝑝 = [𝑧] 𝑝 ∈ ( 𝐴 / )))
5352imp 410 . . . . . . 7 ((𝑧 𝐴𝑝 = [𝑧] ) → 𝑝 ∈ ( 𝐴 / ))
5450, 53syl6 35 . . . . . 6 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → 𝑝 ∈ ( 𝐴 / )))
5554eximdv 1918 . . . . 5 (Prt 𝐴 → (∃𝑧(𝑧𝑝𝑝𝐴) → ∃𝑧 𝑝 ∈ ( 𝐴 / )))
56 19.41v 1950 . . . . 5 (∃𝑧(𝑧𝑝𝑝𝐴) ↔ (∃𝑧 𝑧𝑝𝑝𝐴))
57 19.9v 1988 . . . . 5 (∃𝑧 𝑝 ∈ ( 𝐴 / ) ↔ 𝑝 ∈ ( 𝐴 / ))
5855, 56, 573imtr3g 298 . . . 4 (Prt 𝐴 → ((∃𝑧 𝑧𝑝𝑝𝐴) → 𝑝 ∈ ( 𝐴 / )))
5946, 58syl5 34 . . 3 (Prt 𝐴 → (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ ( 𝐴 / )))
6040, 59impbid 215 . 2 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) ↔ 𝑝 ∈ (𝐴 ∖ {∅})))
6160eqrdv 2796 1 (Prt 𝐴 → ( 𝐴 / ) = (𝐴 ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2111  wne 2987  wral 3106  wrex 3107  Vcvv 3441  cdif 3878  c0 4243  {csn 4525   cuni 4800  {copab 5092  [cec 8270   / cqs 8271  Prt wprt 36164
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ec 8274  df-qs 8278  df-prt 36165
This theorem is referenced by: (None)
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