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Theorem prter2 34689
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 3377 . . . . . . . . . . 11 (∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ))
2 r19.41v 3237 . . . . . . . . . . . 12 (∃𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ) ↔ (∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
32exbii 1924 . . . . . . . . . . 11 (∃𝑧𝑣𝐴 (𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
41, 3bitri 264 . . . . . . . . . 10 (∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
5 df-rex 3067 . . . . . . . . . . 11 (∃𝑧𝑣 𝑝 = [𝑧] ↔ ∃𝑧(𝑧𝑣𝑝 = [𝑧] ))
65rexbii 3189 . . . . . . . . . 10 (∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] ↔ ∃𝑣𝐴𝑧(𝑧𝑣𝑝 = [𝑧] ))
7 vex 3354 . . . . . . . . . . . 12 𝑝 ∈ V
87elqs 7951 . . . . . . . . . . 11 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑧 𝐴𝑝 = [𝑧] )
9 df-rex 3067 . . . . . . . . . . . 12 (∃𝑧 𝐴𝑝 = [𝑧] ↔ ∃𝑧(𝑧 𝐴𝑝 = [𝑧] ))
10 eluni2 4578 . . . . . . . . . . . . . 14 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
1110anbi1i 610 . . . . . . . . . . . . 13 ((𝑧 𝐴𝑝 = [𝑧] ) ↔ (∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
1211exbii 1924 . . . . . . . . . . . 12 (∃𝑧(𝑧 𝐴𝑝 = [𝑧] ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
139, 12bitri 264 . . . . . . . . . . 11 (∃𝑧 𝐴𝑝 = [𝑧] ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
148, 13bitri 264 . . . . . . . . . 10 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑧(∃𝑣𝐴 𝑧𝑣𝑝 = [𝑧] ))
154, 6, 143bitr4ri 293 . . . . . . . . 9 (𝑝 ∈ ( 𝐴 / ) ↔ ∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] )
16 prtlem18.1 . . . . . . . . . . . 12 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
1716prtlem19 34686 . . . . . . . . . . 11 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))
1817ralrimivv 3119 . . . . . . . . . 10 (Prt 𝐴 → ∀𝑣𝐴𝑧𝑣 𝑣 = [𝑧] )
19 2r19.29 3227 . . . . . . . . . . 11 ((∀𝑣𝐴𝑧𝑣 𝑣 = [𝑧] ∧ ∃𝑣𝐴𝑧𝑣 𝑝 = [𝑧] ) → ∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ))
2019ex 397 . . . . . . . . . 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 232 . . . . . . . 8 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] )))
23 eqtr3 2792 . . . . . . . . . 10 ((𝑣 = [𝑧] 𝑝 = [𝑧] ) → 𝑣 = 𝑝)
2423reximi 3159 . . . . . . . . 9 (∃𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ) → ∃𝑧𝑣 𝑣 = 𝑝)
2524reximi 3159 . . . . . . . 8 (∃𝑣𝐴𝑧𝑣 (𝑣 = [𝑧] 𝑝 = [𝑧] ) → ∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝)
2622, 25syl6 35 . . . . . . 7 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝))
27 df-rex 3067 . . . . . . . . . 10 (∃𝑧𝑣 𝑣 = 𝑝 ↔ ∃𝑧(𝑧𝑣𝑣 = 𝑝))
28 19.41v 2029 . . . . . . . . . 10 (∃𝑧(𝑧𝑣𝑣 = 𝑝) ↔ (∃𝑧 𝑧𝑣𝑣 = 𝑝))
2927, 28bitri 264 . . . . . . . . 9 (∃𝑧𝑣 𝑣 = 𝑝 ↔ (∃𝑧 𝑧𝑣𝑣 = 𝑝))
3029simprbi 484 . . . . . . . 8 (∃𝑧𝑣 𝑣 = 𝑝𝑣 = 𝑝)
3130reximi 3159 . . . . . . 7 (∃𝑣𝐴𝑧𝑣 𝑣 = 𝑝 → ∃𝑣𝐴 𝑣 = 𝑝)
3226, 31syl6 35 . . . . . 6 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → ∃𝑣𝐴 𝑣 = 𝑝))
33 risset 3210 . . . . . 6 (𝑝𝐴 ↔ ∃𝑣𝐴 𝑣 = 𝑝)
3432, 33syl6ibr 242 . . . . 5 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝𝐴))
3516prtlem400 34678 . . . . . 6 ¬ ∅ ∈ ( 𝐴 / )
36 nelelne 3041 . . . . . 6 (¬ ∅ ∈ ( 𝐴 / ) → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ≠ ∅))
3735, 36mp1i 13 . . . . 5 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ≠ ∅))
3834, 37jcad 502 . . . 4 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → (𝑝𝐴𝑝 ≠ ∅)))
39 eldifsn 4453 . . . 4 (𝑝 ∈ (𝐴 ∖ {∅}) ↔ (𝑝𝐴𝑝 ≠ ∅))
4038, 39syl6ibr 242 . . 3 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) → 𝑝 ∈ (𝐴 ∖ {∅})))
41 neldifsn 4458 . . . . . . 7 ¬ ∅ ∈ (𝐴 ∖ {∅})
42 n0el 4087 . . . . . . 7 (¬ ∅ ∈ (𝐴 ∖ {∅}) ↔ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧𝑝)
4341, 42mpbi 220 . . . . . 6 𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧𝑝
4443rspec 3080 . . . . 5 (𝑝 ∈ (𝐴 ∖ {∅}) → ∃𝑧 𝑧𝑝)
45 eldifi 3883 . . . . 5 (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝𝐴)
4644, 45jca 501 . . . 4 (𝑝 ∈ (𝐴 ∖ {∅}) → (∃𝑧 𝑧𝑝𝑝𝐴))
4716prtlem19 34686 . . . . . . . . 9 (Prt 𝐴 → ((𝑝𝐴𝑧𝑝) → 𝑝 = [𝑧] ))
4847ancomsd 456 . . . . . . . 8 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → 𝑝 = [𝑧] ))
49 elunii 4579 . . . . . . . 8 ((𝑧𝑝𝑝𝐴) → 𝑧 𝐴)
5048, 49jca2r 34662 . . . . . . 7 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → (𝑧 𝐴𝑝 = [𝑧] )))
51 prtlem11 34674 . . . . . . . . 9 (𝑝 ∈ V → (𝑧 𝐴 → (𝑝 = [𝑧] 𝑝 ∈ ( 𝐴 / ))))
527, 51ax-mp 5 . . . . . . . 8 (𝑧 𝐴 → (𝑝 = [𝑧] 𝑝 ∈ ( 𝐴 / )))
5352imp 393 . . . . . . 7 ((𝑧 𝐴𝑝 = [𝑧] ) → 𝑝 ∈ ( 𝐴 / ))
5450, 53syl6 35 . . . . . 6 (Prt 𝐴 → ((𝑧𝑝𝑝𝐴) → 𝑝 ∈ ( 𝐴 / )))
5554eximdv 1998 . . . . 5 (Prt 𝐴 → (∃𝑧(𝑧𝑝𝑝𝐴) → ∃𝑧 𝑝 ∈ ( 𝐴 / )))
56 19.41v 2029 . . . . 5 (∃𝑧(𝑧𝑝𝑝𝐴) ↔ (∃𝑧 𝑧𝑝𝑝𝐴))
57 19.9v 2065 . . . . 5 (∃𝑧 𝑝 ∈ ( 𝐴 / ) ↔ 𝑝 ∈ ( 𝐴 / ))
5855, 56, 573imtr3g 284 . . . 4 (Prt 𝐴 → ((∃𝑧 𝑧𝑝𝑝𝐴) → 𝑝 ∈ ( 𝐴 / )))
5946, 58syl5 34 . . 3 (Prt 𝐴 → (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ ( 𝐴 / )))
6040, 59impbid 202 . 2 (Prt 𝐴 → (𝑝 ∈ ( 𝐴 / ) ↔ 𝑝 ∈ (𝐴 ∖ {∅})))
6160eqrdv 2769 1 (Prt 𝐴 → ( 𝐴 / ) = (𝐴 ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  c0 4063  {csn 4316   cuni 4574  {copab 4846  [cec 7894   / cqs 7895  Prt wprt 34679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ec 7898  df-qs 7902  df-prt 34680
This theorem is referenced by: (None)
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