Theorem qsel 8379

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 2824	. . 3 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅)
2		eleq2 2904	. . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝐶 ∈ 𝐵))
3		eqeq1 2828	. . . 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 8335	. . . . . 6 ⊢ ((𝐶 ∈ [𝑥]𝑅 ∧ 𝑥 ∈ V) → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶))
6	5	elvd 3503	. . . . 5 ⊢ (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶))
7	6	ibi 269	. . . 4 ⊢ (𝐶 ∈ [𝑥]𝑅 → 𝑥𝑅𝐶)
8		simpll 765	. . . . . 6 ⊢ (((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → 𝑅 Er 𝑋)
9		simpr 487	. . . . . 6 ⊢ (((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
10	8, 9	erthi 8343	. . . . 5 ⊢ (((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
11	10	ex 415	. . . 4 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅))
12	7, 11	syl5 34	. . 3 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅))
13	1, 4, 12	ectocld 8367	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅))
14	13	3impia 1113	1 ⊢ ((𝑅 Er 𝑋 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  Vcvv 3497   class class class wbr 5069   Er wer 8289  [cec 8290   / cqs 8291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-er 8292  df-ec 8294  df-qs 8298

This theorem is referenced by:  frgpnabllem2  18997  prter3  36022