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Theorem ersym 6779

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 6776	. . . . . 6 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
4	2, 3	syl 14	. . . . 5 ⊢ (𝜑 → Rel 𝑅)
5		brrelex12 4788	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	4, 1, 5	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7		brcnvg 4936	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵^◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
8	7	ancoms 268	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵^◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
9	6, 8	syl 14	. . 3 ⊢ (𝜑 → (𝐵^◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
10	1, 9	mpbird 167	. 2 ⊢ (𝜑 → 𝐵^◡𝑅𝐴)
11		df-er 6767	. . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
12	11	simp3bi 1041	. . . . 5 ⊢ (𝑅 Er 𝑋 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
13	2, 12	syl 14	. . . 4 ⊢ (𝜑 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
14	13	unssad 3396	. . 3 ⊢ (𝜑 → ^◡𝑅 ⊆ 𝑅)
15	14	ssbrd 4152	. 2 ⊢ (𝜑 → (𝐵^◡𝑅𝐴 → 𝐵𝑅𝐴))
16	10, 15	mpd 13	1 ⊢ (𝜑 → 𝐵𝑅𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1398 ∈ wcel 2203 Vcvv 2813 ∪ cun 3209 ⊆ wss 3211 class class class wbr 4109 ^◡ccnv 4748 dom cdm 4749 ∘ ccom 4753 Rel wrel 4754 Er wer 6764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-v 2815 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-br 4110 df-opab 4172 df-xp 4755 df-rel 4756 df-cnv 4757 df-er 6767

This theorem is referenced by: ercl2 6780 ersymb 6781 ertr2d 6784 ertr3d 6785 ertr4d 6786 erth 6813 erinxp 6843 qusgrp2 13830 2idlcpblrng 14671

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