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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 8301 | . . . . 5 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
5 | 1, 4 | eleqtrrd 2918 | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
6 | eldmg 5769 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
7 | 1, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
8 | 5, 7 | mpbid 234 | . 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥) |
9 | 2 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝑅 Er 𝑋) |
10 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥) | |
11 | 9, 10, 10 | ertr4d 8310 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴) |
12 | 8, 11 | exlimddv 1936 | 1 ⊢ (𝜑 → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 class class class wbr 5068 dom cdm 5557 Er wer 8288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-er 8291 |
This theorem is referenced by: iserd 8317 erth 8340 iiner 8371 erinxp 8373 nqerid 10357 enqeq 10358 qusgrp 18337 sylow2alem1 18744 sylow2alem2 18745 sylow2a 18746 efginvrel2 18855 efgsrel 18862 efgcpbllemb 18883 frgp0 18888 frgpnabllem1 18995 frgpnabllem2 18996 pcophtb 23635 pi1xfrf 23659 pi1xfr 23661 pi1xfrcnvlem 23662 prtlem10 36003 |
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