![]() |
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 8735 | . . . . 5 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
5 | 1, 4 | eleqtrrd 2828 | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
6 | eldmg 5901 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
7 | 1, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
8 | 5, 7 | mpbid 231 | . 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥) |
9 | 2 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝑅 Er 𝑋) |
10 | simpr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥) | |
11 | 9, 10, 10 | ertr4d 8744 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴) |
12 | 8, 11 | exlimddv 1930 | 1 ⊢ (𝜑 → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 class class class wbr 5149 dom cdm 5678 Er wer 8722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-er 8725 |
This theorem is referenced by: iserd 8751 ecref 8769 erth 8775 iiner 8808 erinxp 8810 nqerid 10958 enqeq 10959 qusgrp 19149 sylow2alem1 19584 sylow2alem2 19585 sylow2a 19586 efginvrel2 19694 efgsrel 19701 efgcpbllemb 19722 frgp0 19727 frgpnabllem1 19840 frgpnabllem2 19841 pcophtb 25000 pi1xfrf 25024 pi1xfr 25026 pi1xfrcnvlem 25027 prtlem10 38467 prjspner01 42184 prjspner1 42185 |
Copyright terms: Public domain | W3C validator |