Theorem elecg 8790

Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
elecg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))

Proof of Theorem elecg

Step	Hyp	Ref	Expression
1		elimasng 6106	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
2	1	ancoms 458	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
3		df-ec 8748	. . 3 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
4	3	eleq2i 2832	. 2 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐴 ∈ (𝑅 “ {𝐵}))
5		df-br 5143	. 2 ⊢ (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
6	2, 4, 5	3bitr4g 314	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2107  {csn 4625  ⟨cop 4631   class class class wbr 5142   “ cima 5687  [cec 8744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748

This theorem is referenced by:  ecref  8791  elec  8792  relelec  8793  ecdmn0  8795  erth  8797  erdisj  8800  qsel  8837  ghmqusnsglem1  19299  ghmquskerlem1  19302  orbsta  19332  sylow2alem1  19636  sylow2blem1  19639  sylow3lem3  19648  efgi2  19744  rngqiprngfulem2  21323  rngqipring1  21327  tgpconncompeqg  24121  xmetec  24445  blpnfctr  24447  xmetresbl  24448  xrsblre  24834  ecxpid  33390  lsmsnorb  33420  ecin0  38354  eqvrelth  38613  qsalrel  42281