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Theorem cnvssb 38726
 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 5527 . 2 (𝐴𝐵𝐴𝐵)
2 cnvss 5527 . . 3 (𝐴𝐵𝐴𝐵)
3 dfrel2 5824 . . . . . . . 8 (Rel 𝐴𝐴 = 𝐴)
43biimpi 208 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
54eqcomd 2831 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
65adantr 474 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴 = 𝐴)
7 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
8 cnvcnvss 5829 . . . . . . 7 𝐵𝐵
97, 8syl6ss 3839 . . . . . 6 (𝐴𝐵𝐴𝐵)
109adantl 475 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
116, 10eqsstrd 3864 . . . 4 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
1211ex 403 . . 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 386   = wceq 1656   ⊆ wss 3798  ◡ccnv 5341  Rel wrel 5347 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350 This theorem is referenced by: (None)
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