Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnvssb Structured version   Visualization version   GIF version

Theorem cnvssb 38675
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 5498 . 2 (𝐴𝐵𝐴𝐵)
2 cnvss 5498 . . 3 (𝐴𝐵𝐴𝐵)
3 dfrel2 5800 . . . . . . . 8 (Rel 𝐴𝐴 = 𝐴)
43biimpi 208 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
54eqcomd 2805 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
65adantr 473 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴 = 𝐴)
7 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
8 cnvcnvss 5805 . . . . . . 7 𝐵𝐵
97, 8syl6ss 3810 . . . . . 6 (𝐴𝐵𝐴𝐵)
109adantl 474 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
116, 10eqsstrd 3835 . . . 4 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
1211ex 402 . . 3 (Rel 𝐴 → (𝐴𝐵𝐴𝐵))
132, 12syl5 34 . 2 (Rel 𝐴 → (𝐴𝐵𝐴𝐵))
141, 13impbid2 218 1 (Rel 𝐴 → (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wss 3769  ccnv 5311  Rel wrel 5317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator