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Theorem cnvssb 42826
Description: Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)
Assertion
Ref Expression
cnvssb (Rel 𝐴 → (𝐴𝐵𝐴𝐵))

Proof of Theorem cnvssb
StepHypRef Expression
1 cnvss 5862 . 2 (𝐴𝐵𝐴𝐵)
2 cnvss 5862 . . 3 (𝐴𝐵𝐴𝐵)
3 dfrel2 6178 . . . . . . . 8 (Rel 𝐴𝐴 = 𝐴)
43biimpi 215 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
54eqcomd 2730 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
65adantr 480 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴 = 𝐴)
7 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
8 cnvcnvss 6183 . . . . . . 7 𝐵𝐵
97, 8sstrdi 3986 . . . . . 6 (𝐴𝐵𝐴𝐵)
109adantl 481 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
116, 10eqsstrd 4012 . . . 4 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
1211ex 412 . . 3 (Rel 𝐴 → (𝐴𝐵𝐴𝐵))
132, 12syl5 34 . 2 (Rel 𝐴 → (𝐴𝐵𝐴𝐵))
141, 13impbid2 225 1 (Rel 𝐴 → (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wss 3940  ccnv 5665  Rel wrel 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674
This theorem is referenced by: (None)
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