Theorem relelec 8052

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 3429	. . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐴 ∈ V)
2		ecexr 8014	. . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)
3	1, 2	jca 507	. . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	3	adantl 475	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ [𝐵]𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5		brrelex12 5389	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
6	5	ancomd 455	. 2 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7		elecg 8050	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
8	4, 6, 7	pm5.21nd 836	1 ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∈ wcel 2164  Vcvv 3414   class class class wbr 4873  Rel wrel 5347  [cec 8007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ec 8011

This theorem is referenced by:  eqgid  17997  tgptsmscls  22323  pstmfval  30473  ismntop  30604  topfneec  32877  releleccnv  34568  elecres  34586  eleccnvep  34590  inecmo  34661  elecxrn  34689  elec1cnvxrn2  34696  eleccossin  34774