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Theorem ersym 6572

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 6569	. . . . . 6 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
4	2, 3	syl 14	. . . . 5 ⊢ (𝜑 → Rel 𝑅)
5		brrelex12 4682	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	4, 1, 5	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7		brcnvg 4826	. . . . 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 6560	. . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
12	11	simp3bi 1016	. . . . 5 ⊢ (𝑅 Er 𝑋 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
13	2, 12	syl 14	. . . 4 ⊢ (𝜑 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
14	13	unssad 3327	. . 3 ⊢ (𝜑 → ^◡𝑅 ⊆ 𝑅)
15	14	ssbrd 4061	. 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 1364 ∈ wcel 2160 Vcvv 2752 ∪ cun 3142 ⊆ wss 3144 class class class wbr 4018 ^◡ccnv 4643 dom cdm 4644 ∘ ccom 4648 Rel wrel 4649 Er wer 6557

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-cnv 4652 df-er 6560

This theorem is referenced by: ercl2 6573 ersymb 6574 ertr2d 6577 ertr3d 6578 ertr4d 6579 erth 6606 erinxp 6636 qusgrp2 13070 2idlcpblrng 13855

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