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Theorem elrelscnveq2 34790
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
elrelscnveq2 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrelscnveq2
StepHypRef Expression
1 cnvsym 5756 . . . 4 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
21a1i 11 . . 3 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
3 cnvsym 5756 . . . . 5 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
4 elrelsrelim 34785 . . . . . . 7 (𝑅 ∈ Rels → Rel 𝑅)
5 dfrel2 5828 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
64, 5sylib 210 . . . . . 6 (𝑅 ∈ Rels → 𝑅 = 𝑅)
76sseq1d 3857 . . . . 5 (𝑅 ∈ Rels → (𝑅𝑅𝑅𝑅))
83, 7syl5rbbr 278 . . . 4 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
9 relbrcnvg 5749 . . . . . . 7 (Rel 𝑅 → (𝑥𝑅𝑦𝑦𝑅𝑥))
104, 9syl 17 . . . . . 6 (𝑅 ∈ Rels → (𝑥𝑅𝑦𝑦𝑅𝑥))
11 relbrcnvg 5749 . . . . . . 7 (Rel 𝑅 → (𝑦𝑅𝑥𝑥𝑅𝑦))
124, 11syl 17 . . . . . 6 (𝑅 ∈ Rels → (𝑦𝑅𝑥𝑥𝑅𝑦))
1310, 12imbi12d 336 . . . . 5 (𝑅 ∈ Rels → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥𝑥𝑅𝑦)))
14132albidv 2022 . . . 4 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
158, 14bitrd 271 . . 3 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
162, 15anbi12d 624 . 2 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦))))
17 eqss 3842 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
18 2albiim 1992 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
1916, 17, 183bitr4g 306 1 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wal 1654   = wceq 1656  wcel 2164  wss 3798   class class class wbr 4875  ccnv 5345  Rel wrel 5351   Rels crels 34525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-cnv 5354  df-rels 34782
This theorem is referenced by:  elrelscnveq4  34791  dfsymrels5  34841
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