Theorem eqvrelqsel 39200

Description: If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 28-Dec-2019.)

Assertion

Ref	Expression
eqvrelqsel	⊢ (( EqvRel 𝑅 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅)

Proof of Theorem eqvrelqsel

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅)
2		eleq2 2852	. . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝐶 ∈ 𝐵))
3		eqeq1 2767	. . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = [𝐶]𝑅 ↔ 𝐵 = [𝐶]𝑅))
4	2, 3	imbi12d 346	. . 3 ⊢ ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅)))
5		elecALTV 38771	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝐶 ∈ [𝑥]𝑅) → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶))
6	5	el2v1 38729	. . . . 5 ⊢ (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶))
7	6	ibi 269	. . . 4 ⊢ (𝐶 ∈ [𝑥]𝑅 → 𝑥𝑅𝐶)
8		simpll 776	. . . . . 6 ⊢ ((( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → EqvRel 𝑅)
9		simpr 488	. . . . . 6 ⊢ ((( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
10	8, 9	eqvrelthi 39197	. . . . 5 ⊢ ((( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
11	10	ex 416	. . . 4 ⊢ (( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅))
12	7, 11	syl5 34	. . 3 ⊢ (( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅))
13	1, 4, 12	ectocld 8765	. 2 ⊢ (( EqvRel 𝑅 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅))
14	13	3impia 1131	1 ⊢ (( EqvRel 𝑅 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  Vcvv 3455   class class class wbr 5101  [cec 8677   / cqs 8678   EqvRel weqvrel 38700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ec 8681  df-qs 8685  df-refrel 39092  df-symrel 39124  df-trrel 39158  df-eqvrel 39169

This theorem is referenced by:  erimeq2  39263