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Theorem cnvssb 43016
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 5875 . 2 (𝐴𝐵𝐴𝐵)
2 cnvss 5875 . . 3 (𝐴𝐵𝐴𝐵)
3 dfrel2 6193 . . . . . . . 8 (Rel 𝐴𝐴 = 𝐴)
43biimpi 215 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
54eqcomd 2734 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
65adantr 480 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴 = 𝐴)
7 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
8 cnvcnvss 6198 . . . . . . 7 𝐵𝐵
97, 8sstrdi 3992 . . . . . 6 (𝐴𝐵𝐴𝐵)
109adantl 481 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
116, 10eqsstrd 4018 . . . 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 1534  wss 3947  ccnv 5677  Rel wrel 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686
This theorem is referenced by: (None)
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