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| 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.) | 
| Ref | Expression | 
|---|---|
| ersymb.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) | 
| erref.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑋) | 
| Ref | Expression | 
|---|---|
| erref | ⊢ (𝜑 → 𝐴𝑅𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | erref.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
| 2 | ersymb.1 | . . . . 5 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
| 3 | erdm 8756 | . . . . 5 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
| 4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑅 = 𝑋) | 
| 5 | 1, 4 | eleqtrrd 2843 | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) | 
| 6 | eldmg 5908 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
| 7 | 1, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | 
| 8 | 5, 7 | mpbid 232 | . 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥) | 
| 9 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝑅 Er 𝑋) | 
| 10 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥) | |
| 11 | 9, 10, 10 | ertr4d 8765 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴) | 
| 12 | 8, 11 | exlimddv 1934 | 1 ⊢ (𝜑 → 𝐴𝑅𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 class class class wbr 5142 dom cdm 5684 Er wer 8743 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-er 8746 | 
| This theorem is referenced by: iserd 8772 ecref 8791 erth 8797 iiner 8830 erinxp 8832 nqerid 10974 enqeq 10975 qusgrp 19205 sylow2alem1 19636 sylow2alem2 19637 sylow2a 19638 efginvrel2 19746 efgsrel 19753 efgcpbllemb 19774 frgp0 19779 frgpnabllem1 19892 frgpnabllem2 19893 pcophtb 25063 pi1xfrf 25087 pi1xfr 25089 pi1xfrcnvlem 25090 prtlem10 38867 prjspner01 42640 prjspner1 42641 | 
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