Theorem erref 6457

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 6447	. . . . 5 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
4	2, 3	syl 14	. . . 4 ⊢ (𝜑 → dom 𝑅 = 𝑋)
5	1, 4	eleqtrrd 2220	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
6		eldmg 4742	. . . 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 6456	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴)
12	8, 11	exlimddv 1871	1 ⊢ (𝜑 → 𝐴𝑅𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481   class class class wbr 3937  dom cdm 4547   Er wer 6434

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-er 6437

This theorem is referenced by:  iserd  6463  erth  6481  iinerm  6509  erinxp  6511