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Theorem eqvrelcoels4 34905
 Description: Two ways to express equivalent coelements. (Contributed by Peter Mazsa, 20-Oct-2021.)
Assertion
Ref Expression
eqvrelcoels4 ( EqvRel ∼ 𝐴 ↔ ∀𝑥𝑧(([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) ≠ ∅ → ([𝑥]( E ↾ 𝐴) ∩ [𝑧]( E ↾ 𝐴)) ≠ ∅))
Distinct variable group:   𝑥,𝐴,𝑧

Proof of Theorem eqvrelcoels4
StepHypRef Expression
1 eqvrelcoss4 34904 . 2 ( EqvRel ≀ ( E ↾ 𝐴) ↔ ∀𝑥𝑧(([𝑥] ≀ ( E ↾ 𝐴) ∩ [𝑧] ≀ ( E ↾ 𝐴)) ≠ ∅ → ([𝑥]( E ↾ 𝐴) ∩ [𝑧]( E ↾ 𝐴)) ≠ ∅))
2 df-coels 34713 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32eqvreleqi 34888 . 2 ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
42eceq2i 34586 . . . . . 6 [𝑥] ∼ 𝐴 = [𝑥] ≀ ( E ↾ 𝐴)
52eceq2i 34586 . . . . . 6 [𝑧] ∼ 𝐴 = [𝑧] ≀ ( E ↾ 𝐴)
64, 5ineq12i 4041 . . . . 5 ([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) = ([𝑥] ≀ ( E ↾ 𝐴) ∩ [𝑧] ≀ ( E ↾ 𝐴))
76neeq1i 3063 . . . 4 (([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) ≠ ∅ ↔ ([𝑥] ≀ ( E ↾ 𝐴) ∩ [𝑧] ≀ ( E ↾ 𝐴)) ≠ ∅)
87imbi1i 341 . . 3 ((([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) ≠ ∅ → ([𝑥]( E ↾ 𝐴) ∩ [𝑧]( E ↾ 𝐴)) ≠ ∅) ↔ (([𝑥] ≀ ( E ↾ 𝐴) ∩ [𝑧] ≀ ( E ↾ 𝐴)) ≠ ∅ → ([𝑥]( E ↾ 𝐴) ∩ [𝑧]( E ↾ 𝐴)) ≠ ∅))
982albii 1919 . 2 (∀𝑥𝑧(([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) ≠ ∅ → ([𝑥]( E ↾ 𝐴) ∩ [𝑧]( E ↾ 𝐴)) ≠ ∅) ↔ ∀𝑥𝑧(([𝑥] ≀ ( E ↾ 𝐴) ∩ [𝑧] ≀ ( E ↾ 𝐴)) ≠ ∅ → ([𝑥]( E ↾ 𝐴) ∩ [𝑧]( E ↾ 𝐴)) ≠ ∅))
101, 3, 93bitr4i 295 1 ( EqvRel ∼ 𝐴 ↔ ∀𝑥𝑧(([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) ≠ ∅ → ([𝑥]( E ↾ 𝐴) ∩ [𝑧]( E ↾ 𝐴)) ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654   ≠ wne 2999   ∩ cin 3797  ∅c0 4146   E cep 5256  ◡ccnv 5345   ↾ cres 5348  [cec 8012   ≀ ccoss 34519   ∼ ccoels 34520   EqvRel weqvrel 34536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ec 8016  df-coss 34712  df-coels 34713  df-refrel 34805  df-symrel 34833  df-trrel 34863  df-eqvrel 34873 This theorem is referenced by: (None)
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