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Theorem prter2 38056
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 3283 . . . . . . . . . . 11 (∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ))
2 r19.41v 3186 . . . . . . . . . . . 12 (∃𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ) ↔ (∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
32exbii 1848 . . . . . . . . . . 11 (∃𝑧𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
41, 3bitri 274 . . . . . . . . . 10 (∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
5 df-rex 3069 . . . . . . . . . . 11 (∃𝑧𝑣 𝑝 = [𝑧] ↔ ∃𝑧(𝑧𝑣𝑝 = [𝑧] ))
65rexbii 3092 . . . . . . . . . 10 (∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] ↔ ∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ))
7 vex 3476 . . . . . . . . . . . 12 𝑝 ∈ V
87elqs 8767 . . . . . . . . . . 11 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑧 𝐴𝑝 = [𝑧] )
9 df-rex 3069 . . . . . . . . . . . 12 (∃𝑧 𝐴𝑝 = [𝑧] ↔ ∃𝑧(𝑧 𝐴𝑝 = [𝑧] ))
10 eluni2 4913 . . . . . . . . . . . . . 14 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
1110anbi1i 622 . . . . . . . . . . . . 13 ((𝑧 𝐴𝑝 = [𝑧] ) ↔ (∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
1211exbii 1848 . . . . . . . . . . . 12 (∃𝑧(𝑧 𝐴𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
139, 12bitri 274 . . . . . . . . . . 11 (∃𝑧 𝐴𝑝 = [𝑧] ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
148, 13bitri 274 . . . . . . . . . 10 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
154, 6, 143bitr4ri 303 . . . . . . . . 9 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] )
16 prtlem18.1 . . . . . . . . . . . 12 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
1716prtlem19 38053 . . . . . . . . . . 11 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))
1817ralrimivv 3196 . . . . . . . . . 10 (Prt 𝐴 → ∀𝑣𝐴𝑧𝑣 𝑣 = [𝑧] )
19 2r19.29 3137 . . . . . . . . . . 11 ((∀𝑣𝐴𝑧𝑣 𝑣 = [𝑧] ∧ ∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] ) → ∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ))
2019ex 411 . . . . . . . . . 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, 21biimtrid 241 . . . . . . . 8 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] )))
23 eqtr3 2756 . . . . . . . . . 10 ((𝑣 = [𝑧] 𝑝 = [𝑧] ) → 𝑣 = 𝑝)
2423reximi 3082 . . . . . . . . 9 (∃𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ) → ∃𝑧𝑣 𝑣 = 𝑝)
2524reximi 3082 . . . . . . . 8 (∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ) → ∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝)
2622, 25syl6 35 . . . . . . 7 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝))
27 df-rex 3069 . . . . . . . . . 10 (∃𝑧𝑣 𝑣 = 𝑝 ↔ ∃𝑧(𝑧𝑣𝑣 = 𝑝))
28 19.41v 1951 . . . . . . . . . 10 (∃𝑧(𝑧𝑣𝑣 = 𝑝) ↔ (∃𝑧 𝑧𝑣𝑣 = 𝑝))
2927, 28bitri 274 . . . . . . . . 9 (∃𝑧𝑣 𝑣 = 𝑝 ↔ (∃𝑧 𝑧𝑣𝑣 = 𝑝))
3029simprbi 495 . . . . . . . 8 (∃𝑧𝑣 𝑣 = 𝑝𝑣 = 𝑝)
3130reximi 3082 . . . . . . 7 (∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝 → ∃𝑣𝐴 𝑣 = 𝑝)
3226, 31syl6 35 . . . . . 6 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴 𝑣 = 𝑝))
33 risset 3228 . . . . . 6 (𝑝𝐴 ↔ ∃𝑣𝐴 𝑣 = 𝑝)
3432, 33imbitrrdi 251 . . . . 5 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝𝐴))
3516prtlem400 38045 . . . . . 6 ¬ ∅ ∈ ( 𝐴 / )
36 nelelne 3039 . . . . . 6 (¬ ∅ ∈ ( 𝐴 / ) → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ≠ ∅))
3735, 36mp1i 13 . . . . 5 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ≠ ∅))
3834, 37jcad 511 . . . 4 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → (𝑝𝐴𝑝 ≠ ∅)))
39 eldifsn 4791 . . . 4 (𝑝 ∈ (𝐴 ∖ {∅}) ↔ (𝑝𝐴𝑝 ≠ ∅))
4038, 39imbitrrdi 251 . . 3 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ∈ (𝐴 ∖ {∅})))
41 neldifsn 4796 . . . . . . 7 ¬ ∅ ∈ (𝐴 ∖ {∅})
42 n0el 4362 . . . . . . 7 (¬ ∅ ∈ (𝐴 ∖ {∅}) ↔ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧𝑝)
4341, 42mpbi 229 . . . . . 6 𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧𝑝
4443rspec 3245 . . . . 5 (𝑝 ∈ (𝐴 ∖ {∅}) → ∃𝑧 𝑧𝑝)
45 eldifi 4127 . . . . 5 (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝𝐴)
4644, 45jca 510 . . . 4 (𝑝 ∈ (𝐴 ∖ {∅}) → (∃𝑧 𝑧𝑝𝑝𝐴))
4716prtlem19 38053 . . . . . . . . 9 (Prt 𝐴 → ((𝑝𝐴𝑧𝑝) → 𝑝 = [𝑧] ))
4847ancomsd 464 . . . . . . . 8 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → 𝑝 = [𝑧] ))
49 elunii 4914 . . . . . . . 8 ((𝑧𝑝𝑝𝐴) → 𝑧 𝐴)
5048, 49jca2r 38030 . . . . . . 7 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → (𝑧 𝐴𝑝 = [𝑧] )))
51 prtlem11 38041 . . . . . . . . 9 (𝑝 ∈ V → (𝑧 𝐴 → (𝑝 = [𝑧] 𝑝 ∈ ( 𝐴 / ))))
5251elv 3478 . . . . . . . 8 (𝑧 𝐴 → (𝑝 = [𝑧] 𝑝 ∈ ( 𝐴 / )))
5352imp 405 . . . . . . 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 1951 . . . . 5 (∃𝑧(𝑧𝑝𝑝𝐴) ↔ (∃𝑧 𝑧𝑝𝑝𝐴))
57 19.9v 1985 . . . . 5 (∃𝑧 𝑝 ∈ ( 𝐴 / ) ↔ 𝑝 ∈ ( 𝐴 / ))
5855, 56, 573imtr3g 294 . . . 4 (Prt 𝐴 → ((∃𝑧 𝑧𝑝𝑝𝐴) → 𝑝 ∈ ( 𝐴 / )))
5946, 58syl5 34 . . 3 (Prt 𝐴 → (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ ( 𝐴 / )))
6040, 59impbid 211 . 2 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) ↔ 𝑝 ∈ (𝐴 ∖ {∅})))
6160eqrdv 2728 1 (Prt 𝐴 → ( 𝐴 / ) = (𝐴 ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wex 1779  wcel 2104  wne 2938  wral 3059  wrex 3068  Vcvv 3472  cdif 3946  c0 4323  {csn 4629   cuni 4909  {copab 5211  [cec 8705   / cqs 8706  Prt wprt 38046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  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-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8709  df-qs 8713  df-prt 38047
This theorem is referenced by: (None)
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