Theorem ercl 8729

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 8727	. . . 4 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → Rel 𝑅)
4		ersym.2	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
5		releldm 5926	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
6	3, 4, 5	syl2anc 596	. 2 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
7		erdm 8728	. . 3 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
8	1, 7	syl 18	. 2 ⊢ (𝜑 → dom 𝑅 = 𝑋)
9	6, 8	eleqtrd 2863	1 ⊢ (𝜑 → 𝐴 ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  dom cdm 5651  Rel wrel 5656   Er wer 8714

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dm 5661  df-er 8717

This theorem is used by:  ercl2  8731  erthi  8774  qliftfun  8823  efgcpbl2  19971  frgpcpbl  19973  prter3  39939