Theorem erref 6533

Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ersymb.1	⊢ (𝜑 → 𝑅 Er 𝑋)
erref.2	⊢ (𝜑 → 𝐴 ∈ 𝑋)

Assertion

Ref	Expression
erref	⊢ (𝜑 → 𝐴𝑅𝐴)

Proof of Theorem erref

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erref.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
2		ersymb.1	. . . . 5 ⊢ (𝜑 → 𝑅 Er 𝑋)
3		erdm 6523	. . . . 5 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
4	2, 3	syl 14	. . . 4 ⊢ (𝜑 → dom 𝑅 = 𝑋)
5	1, 4	eleqtrrd 2250	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
6		eldmg 4806	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
7	1, 6	syl 14	. . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
8	5, 7	mpbid 146	. 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥)
9	2	adantr 274	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝑅 Er 𝑋)
10		simpr 109	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥)
11	9, 10, 10	ertr4d 6532	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴)
12	8, 11	exlimddv 1891	1 ⊢ (𝜑 → 𝐴𝑅𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141   class class class wbr 3989  dom cdm 4611   Er wer 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-er 6513

This theorem is referenced by:  iserd  6539  erth  6557  iinerm  6585  erinxp  6587