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Theorem cnvssb 41865
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 5829 . 2 (𝐴𝐵𝐴𝐵)
2 cnvss 5829 . . 3 (𝐴𝐵𝐴𝐵)
3 dfrel2 6142 . . . . . . . 8 (Rel 𝐴𝐴 = 𝐴)
43biimpi 215 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
54eqcomd 2743 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
65adantr 482 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴 = 𝐴)
7 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
8 cnvcnvss 6147 . . . . . . 7 𝐵𝐵
97, 8sstrdi 3957 . . . . . 6 (𝐴𝐵𝐴𝐵)
109adantl 483 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
116, 10eqsstrd 3983 . . . 4 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
1211ex 414 . . 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 397   = wceq 1542  wss 3911  ccnv 5633  Rel wrel 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642
This theorem is referenced by: (None)
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