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Theorem prter2 33184
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 3198 . . . . . . . . . . 11 (∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ))
2 r19.41v 3070 . . . . . . . . . . . 12 (∃𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ) ↔ (∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
32exbii 1764 . . . . . . . . . . 11 (∃𝑧𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
41, 3bitri 263 . . . . . . . . . 10 (∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
5 df-rex 2902 . . . . . . . . . . 11 (∃𝑧𝑣 𝑝 = [𝑧] ↔ ∃𝑧(𝑧𝑣𝑝 = [𝑧] ))
65rexbii 3023 . . . . . . . . . 10 (∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] ↔ ∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ))
7 vex 3176 . . . . . . . . . . . 12 𝑝 ∈ V
87elqs 7686 . . . . . . . . . . 11 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑧 𝐴𝑝 = [𝑧] )
9 df-rex 2902 . . . . . . . . . . . 12 (∃𝑧 𝐴𝑝 = [𝑧] ↔ ∃𝑧(𝑧 𝐴𝑝 = [𝑧] ))
10 eluni2 4376 . . . . . . . . . . . . . 14 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
1110anbi1i 727 . . . . . . . . . . . . 13 ((𝑧 𝐴𝑝 = [𝑧] ) ↔ (∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
1211exbii 1764 . . . . . . . . . . . 12 (∃𝑧(𝑧 𝐴𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
139, 12bitri 263 . . . . . . . . . . 11 (∃𝑧 𝐴𝑝 = [𝑧] ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
148, 13bitri 263 . . . . . . . . . 10 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
154, 6, 143bitr4ri 292 . . . . . . . . 9 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] )
16 prtlem18.1 . . . . . . . . . . . 12 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
1716prtlem19 33181 . . . . . . . . . . 11 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))
1817ralrimivv 2953 . . . . . . . . . 10 (Prt 𝐴 → ∀𝑣𝐴𝑧𝑣 𝑣 = [𝑧] )
19 2r19.29 33160 . . . . . . . . . . 11 ((∀𝑣𝐴𝑧𝑣 𝑣 = [𝑧] ∧ ∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] ) → ∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ))
2019ex 449 . . . . . . . . . 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 231 . . . . . . . 8 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] )))
23 eqtr3 2631 . . . . . . . . . 10 ((𝑣 = [𝑧] 𝑝 = [𝑧] ) → 𝑣 = 𝑝)
2423reximi 2994 . . . . . . . . 9 (∃𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ) → ∃𝑧𝑣 𝑣 = 𝑝)
2524reximi 2994 . . . . . . . 8 (∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ) → ∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝)
2622, 25syl6 34 . . . . . . 7 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝))
27 df-rex 2902 . . . . . . . . . 10 (∃𝑧𝑣 𝑣 = 𝑝 ↔ ∃𝑧(𝑧𝑣𝑣 = 𝑝))
28 19.41v 1901 . . . . . . . . . 10 (∃𝑧(𝑧𝑣𝑣 = 𝑝) ↔ (∃𝑧 𝑧𝑣𝑣 = 𝑝))
2927, 28bitri 263 . . . . . . . . 9 (∃𝑧𝑣 𝑣 = 𝑝 ↔ (∃𝑧 𝑧𝑣𝑣 = 𝑝))
3029simprbi 479 . . . . . . . 8 (∃𝑧𝑣 𝑣 = 𝑝𝑣 = 𝑝)
3130reximi 2994 . . . . . . 7 (∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝 → ∃𝑣𝐴 𝑣 = 𝑝)
3226, 31syl6 34 . . . . . 6 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴 𝑣 = 𝑝))
33 risset 3044 . . . . . 6 (𝑝𝐴 ↔ ∃𝑣𝐴 𝑣 = 𝑝)
3432, 33syl6ibr 241 . . . . 5 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝𝐴))
3516prtlem400 33173 . . . . . 6 ¬ ∅ ∈ ( 𝐴 / )
36 nelelne 2880 . . . . . 6 (¬ ∅ ∈ ( 𝐴 / ) → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ≠ ∅))
3735, 36mp1i 13 . . . . 5 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ≠ ∅))
3834, 37jcad 554 . . . 4 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → (𝑝𝐴𝑝 ≠ ∅)))
39 eldifsn 4260 . . . 4 (𝑝 ∈ (𝐴 ∖ {∅}) ↔ (𝑝𝐴𝑝 ≠ ∅))
4038, 39syl6ibr 241 . . 3 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ∈ (𝐴 ∖ {∅})))
41 neldifsn 4262 . . . . . . 7 ¬ ∅ ∈ (𝐴 ∖ {∅})
42 n0el 33164 . . . . . . 7 (¬ ∅ ∈ (𝐴 ∖ {∅}) ↔ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧𝑝)
4341, 42mpbi 219 . . . . . 6 𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧𝑝
4443rspec 2915 . . . . 5 (𝑝 ∈ (𝐴 ∖ {∅}) → ∃𝑧 𝑧𝑝)
45 eldifi 3694 . . . . 5 (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝𝐴)
4644, 45jca 553 . . . 4 (𝑝 ∈ (𝐴 ∖ {∅}) → (∃𝑧 𝑧𝑝𝑝𝐴))
4716prtlem19 33181 . . . . . . . . 9 (Prt 𝐴 → ((𝑝𝐴𝑧𝑝) → 𝑝 = [𝑧] ))
4847ancomsd 469 . . . . . . . 8 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → 𝑝 = [𝑧] ))
49 elunii 4377 . . . . . . . 8 ((𝑧𝑝𝑝𝐴) → 𝑧 𝐴)
5048, 49jca2r 33155 . . . . . . 7 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → (𝑧 𝐴𝑝 = [𝑧] )))
51 prtlem11 33169 . . . . . . . . 9 (𝑝 ∈ V → (𝑧 𝐴 → (𝑝 = [𝑧] 𝑝 ∈ ( 𝐴 / ))))
527, 51ax-mp 5 . . . . . . . 8 (𝑧 𝐴 → (𝑝 = [𝑧] 𝑝 ∈ ( 𝐴 / )))
5352imp 444 . . . . . . 7 ((𝑧 𝐴𝑝 = [𝑧] ) → 𝑝 ∈ ( 𝐴 / ))
5450, 53syl6 34 . . . . . 6 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → 𝑝 ∈ ( 𝐴 / )))
5554eximdv 1833 . . . . 5 (Prt 𝐴 → (∃𝑧(𝑧𝑝𝑝𝐴) → ∃𝑧 𝑝 ∈ ( 𝐴 / )))
56 19.41v 1901 . . . . 5 (∃𝑧(𝑧𝑝𝑝𝐴) ↔ (∃𝑧 𝑧𝑝𝑝𝐴))
57 19.9v 1883 . . . . 5 (∃𝑧 𝑝 ∈ ( 𝐴 / ) ↔ 𝑝 ∈ ( 𝐴 / ))
5855, 56, 573imtr3g 283 . . . 4 (Prt 𝐴 → ((∃𝑧 𝑧𝑝𝑝𝐴) → 𝑝 ∈ ( 𝐴 / )))
5946, 58syl5 33 . . 3 (Prt 𝐴 → (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ ( 𝐴 / )))
6040, 59impbid 201 . 2 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) ↔ 𝑝 ∈ (𝐴 ∖ {∅})))
6160eqrdv 2608 1 (Prt 𝐴 → ( 𝐴 / ) = (𝐴 ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wex 1695  wcel 1977  wne 2780  wral 2896  wrex 2897  Vcvv 3173  cdif 3537  c0 3874  {csn 4125   cuni 4372  {copab 4642  [cec 7627   / cqs 7628  Prt wprt 33174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ec 7631  df-qs 7635  df-prt 33175
This theorem is referenced by: (None)
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