Theorem qsel 6499

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 2137	. . 3 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅)
2		eleq2 2201	. . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝐶 ∈ 𝐵))
3		eqeq1 2144	. . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = [𝐶]𝑅 ↔ 𝐵 = [𝐶]𝑅))
4	2, 3	imbi12d 233	. . 3 ⊢ ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅)))
5		vex 2684	. . . . . 6 ⊢ 𝑥 ∈ V
6		elecg 6460	. . . . . 6 ⊢ ((𝐶 ∈ [𝑥]𝑅 ∧ 𝑥 ∈ V) → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶))
7	5, 6	mpan2 421	. . . . 5 ⊢ (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶))
8	7	ibi 175	. . . 4 ⊢ (𝐶 ∈ [𝑥]𝑅 → 𝑥𝑅𝐶)
9		simpll 518	. . . . . 6 ⊢ (((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → 𝑅 Er 𝑋)
10		simpr 109	. . . . . 6 ⊢ (((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
11	9, 10	erthi 6468	. . . . 5 ⊢ (((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
12	11	ex 114	. . . 4 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅))
13	8, 12	syl5 32	. . 3 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅))
14	1, 4, 13	ectocld 6488	. 2 ⊢ ((𝑅 Er 𝑋 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅))
15	14	3impia 1178	1 ⊢ ((𝑅 Er 𝑋 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  Vcvv 2681   class class class wbr 3924   Er wer 6419  [cec 6420   / cqs 6421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-er 6422  df-ec 6424  df-qs 6428

This theorem is referenced by: (None)