Theorem ercl 8303

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 8301	. . . 4 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → Rel 𝑅)
4		ersym.2	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
5		releldm 5817	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
6	3, 4, 5	syl2anc 586	. 2 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
7		erdm 8302	. . 3 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
8	1, 7	syl 17	. 2 ⊢ (𝜑 → dom 𝑅 = 𝑋)
9	6, 8	eleqtrd 2918	1 ⊢ (𝜑 → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113   class class class wbr 5069  dom cdm 5558  Rel wrel 5563   Er wer 8289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-dm 5568  df-er 8292

This theorem is referenced by:  ercl2  8305  erthi  8343  qliftfun  8385  efgcpbl2  18886  frgpcpbl  18888  prter3  36022