Theorem ercl 8708

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 8706	. . . 4 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → Rel 𝑅)
4		ersym.2	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
5		releldm 5937	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
6	3, 4, 5	syl2anc 595	. 2 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
7		erdm 8707	. . 3 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
8	1, 7	syl 18	. 2 ⊢ (𝜑 → dom 𝑅 = 𝑋)
9	6, 8	eleqtrd 2871	1 ⊢ (𝜑 → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149   class class class wbr 5113  dom cdm 5664  Rel wrel 5669   Er wer 8693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5407

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5670  df-rel 5671  df-dm 5674  df-er 8696

This theorem is referenced by:  ercl2  8710  erthi  8753  qliftfun  8802  efgcpbl2  19829  frgpcpbl  19831  prter3  39583