Theorem elecg 8768

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 6093	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
2	1	ancoms 457	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
3		df-ec 8727	. . 3 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
4	3	eleq2i 2817	. 2 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐴 ∈ (𝑅 “ {𝐵}))
5		df-br 5150	. 2 ⊢ (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
6	2, 4, 5	3bitr4g 313	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∈ wcel 2098  {csn 4630  ⟨cop 4636   class class class wbr 5149   “ cima 5681  [cec 8723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727

This theorem is referenced by:  ecref  8769  elec  8770  relelec  8771  ecdmn0  8773  erth  8775  erdisj  8778  qsel  8815  ghmqusnsglem1  19243  ghmquskerlem1  19246  orbsta  19276  sylow2alem1  19584  sylow2blem1  19587  sylow3lem3  19596  efgi2  19692  rngqiprngfulem2  21219  rngqipring1  21223  tgpconncompeqg  24060  xmetec  24384  blpnfctr  24386  xmetresbl  24387  xrsblre  24771  ecxpid  33172  lsmsnorb  33203  ecin0  37954  eqvrelth  38213  qsalrel  41864