Theorem ecelqsg 5980

Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.)

Assertion

Ref	Expression
ecelqsg	⊢ ((R ∈ V ∧ B ∈ A) → [B]R ∈ (A / R))

Proof of Theorem ecelqsg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2353	. . 3 ⊢ [B]R = [B]R
2		eceq1 5963	. . . . 5 ⊢ (x = B → [x]R = [B]R)
3	2	eqeq2d 2364	. . . 4 ⊢ (x = B → ([B]R = [x]R ↔ [B]R = [B]R))
4	3	rspcev 2956	. . 3 ⊢ ((B ∈ A ∧ [B]R = [B]R) → ∃x ∈ A [B]R = [x]R)
5	1, 4	mpan2 652	. 2 ⊢ (B ∈ A → ∃x ∈ A [B]R = [x]R)
6		ecexg 5950	. . . 4 ⊢ (R ∈ V → [B]R ∈ V)
7		elqsg 5977	. . . 4 ⊢ ([B]R ∈ V → ([B]R ∈ (A / R) ↔ ∃x ∈ A [B]R = [x]R))
8	6, 7	syl 15	. . 3 ⊢ (R ∈ V → ([B]R ∈ (A / R) ↔ ∃x ∈ A [B]R = [x]R))
9	8	biimpar 471	. 2 ⊢ ((R ∈ V ∧ ∃x ∈ A [B]R = [x]R) → [B]R ∈ (A / R))
10	5, 9	sylan2 460	1 ⊢ ((R ∈ V ∧ B ∈ A) → [B]R ∈ (A / R))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  [cec 5946   / cqs 5947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-ima 4728  df-ec 5948  df-qs 5952

This theorem is referenced by:  ecelqsi  5981