Theorem ercl 8649

Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ersym.1	⊢ (𝜑 → 𝑅 Er 𝑋)
ersym.2	⊢ (𝜑 → 𝐴𝑅𝐵)

Assertion

Ref	Expression
ercl	⊢ (𝜑 → 𝐴 ∈ 𝑋)

Proof of Theorem ercl

Step	Hyp	Ref	Expression
1		ersym.1	. . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋)
2		errel 8647	. . . 4 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → Rel 𝑅)
4		ersym.2	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
5		releldm 5893	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
6	3, 4, 5	syl2anc 591	. 2 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
7		erdm 8648	. . 3 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
8	1, 7	syl 17	. 2 ⊢ (𝜑 → dom 𝑅 = 𝑋)
9	6, 8	eleqtrd 2843	1 ⊢ (𝜑 → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121   class class class wbr 5075  dom cdm 5621  Rel wrel 5626   Er wer 8634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-dm 5631  df-er 8637

This theorem is referenced by:  ercl2  8651  erthi  8694  qliftfun  8743  efgcpbl2  19727  frgpcpbl  19729  prter3  39389