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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 7639 | . . . . 5 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
5 | 1, 4 | eleqtrrd 2691 | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
6 | eldmg 5241 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
7 | 1, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
8 | 5, 7 | mpbid 221 | . 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥) |
9 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝑅 Er 𝑋) |
10 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥) | |
11 | 9, 10, 10 | ertr4d 7648 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴) |
12 | 8, 11 | exlimddv 1850 | 1 ⊢ (𝜑 → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 class class class wbr 4583 dom cdm 5038 Er wer 7626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-er 7629 |
This theorem is referenced by: iserd 7655 erth 7678 iiner 7706 erinxp 7708 nqerid 9634 enqeq 9635 qusgrp 17472 sylow2alem1 17855 sylow2alem2 17856 sylow2a 17857 efginvrel2 17963 efgsrel 17970 efgcpbllemb 17991 frgp0 17996 frgpnabllem1 18099 frgpnabllem2 18100 pcophtb 22637 pi1xfrf 22661 pi1xfr 22663 pi1xfrcnvlem 22664 prtlem10 33168 |
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