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Theorem relcnveq3 38322
Description: Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)
Assertion
Ref Expression
relcnveq3 (Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem relcnveq3
StepHypRef Expression
1 eqss 3999 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
2 cnvsym 6132 . . . . . . 7 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
32biimpi 216 . . . . . 6 (𝑅𝑅 → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
43a1d 25 . . . . 5 (𝑅𝑅 → (Rel 𝑅 → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
54adantl 481 . . . 4 ((𝑅𝑅𝑅𝑅) → (Rel 𝑅 → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
65com12 32 . . 3 (Rel 𝑅 → ((𝑅𝑅𝑅𝑅) → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
7 dfrel2 6209 . . . . 5 (Rel 𝑅𝑅 = 𝑅)
8 cnvss 5883 . . . . . . . 8 (𝑅𝑅𝑅𝑅)
9 sseq1 4009 . . . . . . . 8 (𝑅 = 𝑅 → (𝑅𝑅𝑅𝑅))
108, 9syl5ibcom 245 . . . . . . 7 (𝑅𝑅 → (𝑅 = 𝑅𝑅𝑅))
112, 10sylbir 235 . . . . . 6 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (𝑅 = 𝑅𝑅𝑅))
1211com12 32 . . . . 5 (𝑅 = 𝑅 → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅))
137, 12sylbi 217 . . . 4 (Rel 𝑅 → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅))
142biimpri 228 . . . 4 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅)
1513, 14jca2 513 . . 3 (Rel 𝑅 → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (𝑅𝑅𝑅𝑅)))
166, 15impbid 212 . 2 (Rel 𝑅 → ((𝑅𝑅𝑅𝑅) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
171, 16bitrid 283 1 (Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wss 3951   class class class wbr 5143  ccnv 5684  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by:  relcnveq  38323
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