Theorem relelec 8669

Description: Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)

Assertion

Ref	Expression
relelec	⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))

Proof of Theorem relelec

Step	Hyp	Ref	Expression
1		elex 3457	. . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐴 ∈ V)
2		ecexr 8627	. . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)
3	1, 2	jca 511	. . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	3	adantl 481	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ [𝐵]𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5		brrelex12 5668	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
6	5	ancomd 461	. 2 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7		elecg 8666	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
8	4, 6, 7	pm5.21nd 801	1 ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  Vcvv 3436   class class class wbr 5091  Rel wrel 5621  [cec 8620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ec 8624

This theorem is referenced by:  elecres  8670  eqgid  19090  eqg0el  19093  tgptsmscls  24063  pstmfval  33904  ismntop  34034  topfneec  36388  releleccnv  38291  eleccnvep  38314  inecmo  38382  elecxrn  38417  elec1cnvxrn2  38428  eleccossin  38519