Theorem elecg 8807

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 6118	. . 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 8765	. . 3 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
4	3	eleq2i 2836	. 2 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐴 ∈ (𝑅 “ {𝐵}))
5		df-br 5167	. 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 2108  {csn 4648  ⟨cop 4654   class class class wbr 5166   “ cima 5703  [cec 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765

This theorem is referenced by:  ecref  8808  elec  8809  relelec  8810  ecdmn0  8812  erth  8814  erdisj  8817  qsel  8854  ghmqusnsglem1  19320  ghmquskerlem1  19323  orbsta  19353  sylow2alem1  19659  sylow2blem1  19662  sylow3lem3  19671  efgi2  19767  rngqiprngfulem2  21345  rngqipring1  21349  tgpconncompeqg  24141  xmetec  24465  blpnfctr  24467  xmetresbl  24468  xrsblre  24852  ecxpid  33354  lsmsnorb  33384  ecin0  38308  eqvrelth  38567  qsalrel  42235