Theorem qsel 8828

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.)

Assertion

Ref	Expression
qsel	⊢ ((𝑅 Er 𝑋 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅)

Proof of Theorem qsel

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 347	. . 3 ⊢ ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅)))
5		elecg 8773	. . . . . 6 ⊢ ((𝐶 ∈ [𝑥]𝑅 ∧ 𝑥 ∈ V) → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶))
6	5	elvd 3459	. . . . 5 ⊢ (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶))
7	6	ibi 270	. . . 4 ⊢ (𝐶 ∈ [𝑥]𝑅 → 𝑥𝑅𝐶)
8		simpll 779	. . . . . 6 ⊢ (((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → 𝑅 Er 𝑋)
9		simpr 490	. . . . . 6 ⊢ (((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
10	8, 9	erthi 8785	. . . . 5 ⊢ (((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
11	10	ex 418	. . . 4 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅))
12	7, 11	syl5 35	. . 3 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅))
13	1, 4, 12	ectocld 8814	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅))
14	13	3impia 1135	1 ⊢ ((𝑅 Er 𝑋 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  Vcvv 3453   class class class wbr 5105   Er wer 8725  [cec 8726   / cqs 8727

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-er 8728  df-ec 8730  df-qs 8734

This theorem is used by:  ghmqusnsg  19508  ghmquskerlem3  19512  ghmqusker  19513  frgpnabllem2  20100  rhmqusnsg  21593  lmhmqusker  34021  rhmquskerlem  34028  prter3  40079