ersym - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ersym		Structured version Visualization version GIF version

Theorem ersym 8715

Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

**Hypotheses**
Ref	Expression
ersym.1	⊢ (𝜑 → 𝑅 Er 𝑋)
ersym.2	⊢ (𝜑 → 𝐴𝑅𝐵)

**Assertion**
Ref	Expression
ersym	⊢ (𝜑 → 𝐵𝑅𝐴)

**Proof of Theorem ersym**
Step	Hyp	Ref	Expression
1		ersym.2	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		ersym.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Er 𝑋)
3		errel 8712	. . . . . 6 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → Rel 𝑅)
5		brrelex12 5729	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	4, 1, 5	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7		brcnvg 5880	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵^◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
8	7	ancoms 460	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵^◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
9	6, 8	syl 17	. . 3 ⊢ (𝜑 → (𝐵^◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
10	1, 9	mpbird 257	. 2 ⊢ (𝜑 → 𝐵^◡𝑅𝐴)
11		df-er 8703	. . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
12	11	simp3bi 1148	. . . . 5 ⊢ (𝑅 Er 𝑋 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
13	2, 12	syl 17	. . . 4 ⊢ (𝜑 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
14	13	unssad 4188	. . 3 ⊢ (𝜑 → ^◡𝑅 ⊆ 𝑅)
15	14	ssbrd 5192	. 2 ⊢ (𝜑 → (𝐵^◡𝑅𝐴 → 𝐵𝑅𝐴))
16	10, 15	mpd 15	1 ⊢ (𝜑 → 𝐵𝑅𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3947 ⊆ wss 3949 class class class wbr 5149 ^◡ccnv 5676 dom cdm 5677 ∘ ccom 5681 Rel wrel 5682 Er wer 8700

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-er 8703

This theorem is referenced by: ercl2 8716 ersymb 8717 ertr2d 8720 ertr3d 8721 ertr4d 8722 erth 8752 erinxp 8785 nqereu 10924 nqerf 10925 1nqenq 10957 qusgrp2 18941 efginvrel2 19595 efgcpbllemb 19623 2idlcpbl 20871 tgptsmscls 23654 nsgqusf1olem3 32526 qsnzr 32574 qsalrel 41062 prjspner01 41367 2idlcpblrng 46766

W3C validator