![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > erref | Structured version Visualization version GIF version |
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 8036 | . . . . 5 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
5 | 1, 4 | eleqtrrd 2861 | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
6 | eldmg 5564 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
7 | 1, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
8 | 5, 7 | mpbid 224 | . 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥) |
9 | 2 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝑅 Er 𝑋) |
10 | simpr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥) | |
11 | 9, 10, 10 | ertr4d 8045 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴) |
12 | 8, 11 | exlimddv 1978 | 1 ⊢ (𝜑 → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∃wex 1823 ∈ wcel 2106 class class class wbr 4886 dom cdm 5355 Er wer 8023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4887 df-opab 4949 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-er 8026 |
This theorem is referenced by: iserd 8052 erth 8073 iiner 8102 erinxp 8104 nqerid 10090 enqeq 10091 qusgrp 18033 sylow2alem1 18416 sylow2alem2 18417 sylow2a 18418 efginvrel2 18524 efgsrel 18531 efgcpbllemb 18554 frgp0 18559 frgpnabllem1 18662 frgpnabllem2 18663 pcophtb 23236 pi1xfrf 23260 pi1xfr 23262 pi1xfrcnvlem 23263 prtlem10 35013 |
Copyright terms: Public domain | W3C validator |