Theorem cnvcnvss 6051

Description: The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)

Assertion

Ref	Expression
cnvcnvss	⊢ ◡◡𝐴 ⊆ 𝐴

Proof of Theorem cnvcnvss

Step	Hyp	Ref	Expression
1		cnvcnv 6049	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		inss1 4205	. 2 ⊢ (𝐴 ∩ (V × V)) ⊆ 𝐴
3	1, 2	eqsstri 4001	1 ⊢ ◡◡𝐴 ⊆ 𝐴

Syntax hints:  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936   × cxp 5553  ^◡ccnv 5554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-cnv 5563

This theorem is referenced by:  funcnvcnv  6421  foimacnv  6632  cnvct  8586  cnvfi  8806  structcnvcnv  16497  mvdco  18573  fcoinver  30357  fcnvgreu  30418  cnvssb  39966  relnonrel  39967  clcnvlem  40003  cnvtrrel  40035  relexpaddss  40083