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Theorem relcnveq3 36773
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 3959 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
2 cnvsym 6066 . . . . . . 7 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
32biimpi 215 . . . . . 6 (𝑅𝑅 → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
43a1d 25 . . . . 5 (𝑅𝑅 → (Rel 𝑅 → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
54adantl 482 . . . 4 ((𝑅𝑅𝑅𝑅) → (Rel 𝑅 → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
65com12 32 . . 3 (Rel 𝑅 → ((𝑅𝑅𝑅𝑅) → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
7 dfrel2 6141 . . . . 5 (Rel 𝑅𝑅 = 𝑅)
8 cnvss 5828 . . . . . . . 8 (𝑅𝑅𝑅𝑅)
9 sseq1 3969 . . . . . . . 8 (𝑅 = 𝑅 → (𝑅𝑅𝑅𝑅))
108, 9syl5ibcom 244 . . . . . . 7 (𝑅𝑅 → (𝑅 = 𝑅𝑅𝑅))
112, 10sylbir 234 . . . . . 6 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (𝑅 = 𝑅𝑅𝑅))
1211com12 32 . . . . 5 (𝑅 = 𝑅 → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅))
137, 12sylbi 216 . . . 4 (Rel 𝑅 → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅))
142biimpri 227 . . . 4 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅)
1513, 14jca2 514 . . 3 (Rel 𝑅 → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (𝑅𝑅𝑅𝑅)))
166, 15impbid 211 . 2 (Rel 𝑅 → ((𝑅𝑅𝑅𝑅) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
171, 16bitrid 282 1 (Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wss 3910   class class class wbr 5105  ccnv 5632  Rel wrel 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641
This theorem is referenced by:  relcnveq  36774
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