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Theorem ersym 8775

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 8772	. . . . . 6 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → Rel 𝑅)
5		brrelex12 5752	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	4, 1, 5	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7		brcnvg 5904	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵^◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
8	7	ancoms 458	. . . 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 8763	. . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
12	11	simp3bi 1147	. . . . 5 ⊢ (𝑅 Er 𝑋 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
13	2, 12	syl 17	. . . 4 ⊢ (𝜑 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
14	13	unssad 4216	. . 3 ⊢ (𝜑 → ^◡𝑅 ⊆ 𝑅)
15	14	ssbrd 5209	. 2 ⊢ (𝜑 → (𝐵^◡𝑅𝐴 → 𝐵𝑅𝐴))
16	10, 15	mpd 15	1 ⊢ (𝜑 → 𝐵𝑅𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ∪ cun 3974 ⊆ wss 3976 class class class wbr 5166 ^◡ccnv 5699 dom cdm 5700 ∘ ccom 5704 Rel wrel 5705 Er wer 8760

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-xp 5706 df-rel 5707 df-cnv 5708 df-er 8763

This theorem is referenced by: ercl2 8776 ersymb 8777 ertr2d 8780 ertr3d 8781 ertr4d 8782 erth 8814 erinxp 8849 nqereu 10998 nqerf 10999 1nqenq 11031 qusgrp2 19098 efginvrel2 19769 efgcpbllemb 19797 2idlcpblrng 21304 tgptsmscls 24179 nsgqusf1olem3 33408 qsnzr 33448 qsalrel 42235 prjspner01 42580

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