Theorem eqvrelcoels4 34905

Description: Two ways to express equivalent coelements. (Contributed by Peter Mazsa, 20-Oct-2021.)

Assertion

Ref	Expression
eqvrelcoels4	⊢ ( EqvRel ∼ 𝐴 ↔ ∀𝑥∀𝑧(([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) ≠ ∅ → ([𝑥]◡(◡ E ↾ 𝐴) ∩ [𝑧]◡(◡ E ↾ 𝐴)) ≠ ∅))

Distinct variable group:   𝑥,𝐴,𝑧

Proof of Theorem eqvrelcoels4

Step	Hyp	Ref	Expression
1		eqvrelcoss4 34904	. 2 ⊢ ( EqvRel ≀ (◡ E ↾ 𝐴) ↔ ∀𝑥∀𝑧(([𝑥] ≀ (◡ E ↾ 𝐴) ∩ [𝑧] ≀ (◡ E ↾ 𝐴)) ≠ ∅ → ([𝑥]◡(◡ E ↾ 𝐴) ∩ [𝑧]◡(◡ E ↾ 𝐴)) ≠ ∅))
2		df-coels 34713	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	eqvreleqi 34888	. 2 ⊢ ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	2	eceq2i 34586	. . . . . 6 ⊢ [𝑥] ∼ 𝐴 = [𝑥] ≀ (◡ E ↾ 𝐴)
5	2	eceq2i 34586	. . . . . 6 ⊢ [𝑧] ∼ 𝐴 = [𝑧] ≀ (◡ E ↾ 𝐴)
6	4, 5	ineq12i 4041	. . . . 5 ⊢ ([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) = ([𝑥] ≀ (◡ E ↾ 𝐴) ∩ [𝑧] ≀ (◡ E ↾ 𝐴))
7	6	neeq1i 3063	. . . 4 ⊢ (([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) ≠ ∅ ↔ ([𝑥] ≀ (◡ E ↾ 𝐴) ∩ [𝑧] ≀ (◡ E ↾ 𝐴)) ≠ ∅)
8	7	imbi1i 341	. . 3 ⊢ ((([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) ≠ ∅ → ([𝑥]◡(◡ E ↾ 𝐴) ∩ [𝑧]◡(◡ E ↾ 𝐴)) ≠ ∅) ↔ (([𝑥] ≀ (◡ E ↾ 𝐴) ∩ [𝑧] ≀ (◡ E ↾ 𝐴)) ≠ ∅ → ([𝑥]◡(◡ E ↾ 𝐴) ∩ [𝑧]◡(◡ E ↾ 𝐴)) ≠ ∅))
9	8	2albii 1919	. 2 ⊢ (∀𝑥∀𝑧(([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) ≠ ∅ → ([𝑥]◡(◡ E ↾ 𝐴) ∩ [𝑧]◡(◡ E ↾ 𝐴)) ≠ ∅) ↔ ∀𝑥∀𝑧(([𝑥] ≀ (◡ E ↾ 𝐴) ∩ [𝑧] ≀ (◡ E ↾ 𝐴)) ≠ ∅ → ([𝑥]◡(◡ E ↾ 𝐴) ∩ [𝑧]◡(◡ E ↾ 𝐴)) ≠ ∅))
10	1, 3, 9	3bitr4i 295	1 ⊢ ( EqvRel ∼ 𝐴 ↔ ∀𝑥∀𝑧(([𝑥] ∼ 𝐴 ∩ [𝑧] ∼ 𝐴) ≠ ∅ → ([𝑥]◡(◡ E ↾ 𝐴) ∩ [𝑧]◡(◡ E ↾ 𝐴)) ≠ ∅))

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)